Labor Secretary Thomas Perez, who is said to be on the short list of Hillary Clinton’s running mate, addresses a women’s summit at the Labor Department in June. (Bill O’Leary/The Washington Post)

Just before the United flight took off, the pilot needed to balance the jet’s weight and asked for volunteers to move to the back. So Labor Secretary Thomas Perez stopped writing his speech, packed up his papers and moved to the cramped last row.

As he walked the length of the plane, no other passenger recognized the Cabinet member, who has quietly gained stature in the Obama administration and is widely expected to hold a top job in a Hillary Clinton administration; his name is even being raised as a possible candidate for vice president.

Perez, the Spanish-speaking son of Dominican immigrants, is the least-known name on a list of vice-presidential contenders that includes Sens. Timothy M. Kaine (D-Va.), Elizabeth Warren (D-Mass.) and Cory Booker (D-N.J.).

But he is increasingly showing up on television and in swing states to stump for Clinton.

“Donald Trump is a fraud,” Perez said Sunday on NBC’s “Meet the Press.” “Listening to him talk about how he is going to put America first again — he’s spent his entire career putting his profits first.”

Perez hammered Trump, calling him a “train wreck” and “outsourcer in chief,” and contrasted the Republican candidate, whom he described as “volatile,” with Clinton: “What I have seen working with Secretary Clinton is she is a steady hand.”

One of the reasons Perez is not widely recognized is that the highest elected office he has held was in Maryland as president of the Montgomery County Council — and that was a decade ago. He has long worked in government, including as a federal prosecutor, as counsel to Sen. Edward M. Kennedy, for the state of Maryland and as a leading civil rights attorney.

“He is not a showboater, not flash, an everyday guy,” said Michael Steele, the former head of the Republican National Committee. Steele, who was lieutenant governor of Maryland when Perez was on the council, said Perez could help smooth Clinton’s “strained relationship” with many Republicans.

“He is not an I-am-right, you-are-wrong guy,” Steele said.

On the flight to Cleveland, Taylor Bush said he did not recognize the balding, bespectacled passenger sitting directly behind him. When told Perez was in President Obama’s Cabinet and in the running for an even bigger job in a potential Clinton White House, Bush thought for a moment and said that in the current political climate, “I think it’s kind of an advantage not to be a celebrity. Just being an everyday guy would have appeal.”

Perez is the sleeper in Obama’s Cabinet. He was picked to run the Labor Department three years ago after heading the civil rights division at the Justice Department. There, he investigated a record number of police departments for allegations of using excessive force, as well as the Trayvon Martin shooting in Florida.

At the Labor Department, he has a bigger microphone, and he has used it to advocate for a progressive agenda at a time of growing worry over income equality. In city after city, Perez lobbies for a higher minimum wage: “No one who works a full-time job in this country should have to live in poverty.” He says not enacting laws to give paid leave to parents of newborns, especially poor single mothers, is “unconscionable.”

A big recent win for Perez is a new federal rule that lifts the amount of money middle-class workers can earn and still be eligible for overtime pay. Managers earning even $30,000 are not entitled to get extra pay when they work more than 40 hours a week. On Dec. 1, that changes.

Some Democrats say Perez would help Clinton attract many of those who supported her rival, Sen. Bernie Sanders of Vermont.

But Perez does not make everyone happy.

“Our members are frustrated with Secretary Perez,” said Jack Mozloom, a spokesman for the National Federation of Independent Business, which represents 325,000 businesses with fewer than 10 employees. “He has been very aggressive with the overtime rule,” which Moz­loom thinks will lead to employers to cut back hours. “He too frequently gives the labor community whatever they want.”

Rush Limbaugh, the conservative radio talk-show host, has even likened Perez to socialist Hugo Chávez.

The power of ‘we’

Perez was far more awake than anyone else on the plane when it landed in Ohio at 7 a.m. that day. He does not drink coffee in the morning — doesn’t need it. The lawyer who ran three Boston Marathons before his right knee blew out is already talking about “a fierce urgency” to get things done.

When he walks onstage in a huge Cleveland hotel ballroom, Sen. Sherrod Brown (D-Ohio), also part of the vice-presidential sweepstakes, is on deck to talk to the BlueGreen Alliance, a group of labor unions and environmental organizations.

Perez talks about growing up in Buffalo, just across Lake Erie, and how optimistic he is about the future of the Rust Belt. He has seen abandoned steel mills retooled to turn out solar panels. He promises the government will help “up-skill” workers — and he makes sure to include coal workers.

Coal miners are still smarting from remarks Clinton made earlier this year. When she was talking about the move to cleaner energy, she offered up this sound bite for which she later apologized: “We’re going to put a lot of coal miners and coal companies out of business.”

Perez is winning nods and applause as he keeps talking about the “collective power of ‘we’ ” and how the Labor Department wants to be a kind of “Match.com” linking workers to good jobs.

In constant motion all day, he talks to the utility-union boss, the tugboat welder, the hospital manager. Then, he drives to Newburgh Heights, a little town that just made headlines. Its full-time public employees are now eligible for six months of paid leave after the birth of a child.

Perez pulls up to a roadside tavern, where he meets the mayor to laud the town’s initiative. With two local TV crews filming, Perez says too many women are torn by guilt when they go back to work right after the birth of their child. “They feel like a bad employee and a bad mom.” Employees treated well stay, and although paid leave costs money, “you can’t put a price tag on the benefit of a stable workforce,” he says.

‘We’re vetting VP candidates!’

It is awkward to audition for vice president. Just ask former senator Evan Bayh of Indiana, twice vetted for the job, by Al Gore and Bill Clinton — and twice passed over. It is a rare political job for which you cannot seem to be campaigning. “You are used to being aggressive. You campaign actively, but that is not how the vice-presidential selection works,” Bayh said. “Normal rules don’t apply.”

The selection comes down to a candidate’s calculations about what a running mate brings to the ticket, whether that is geographical balance, personality or other factors.

That makes the veepstakes the biggest guessing game in politics these days.

And for Perez and the others whose names are being tossed around, that means it is best to deflect questions.

“I leave that to others,” he says during a car ride in Cleveland. “I just do my day job.”

Perez, 54, is often compared with Julián Castro, 41, the secretary of housing and urban development, whose name is also being floated as a vice-presidential candidate. Both are Hispanic (Perez speaks fluent Spanish, Castro does not). Both are in Obama’s Cabinet. Both earned their law degrees at Harvard.

Castro is younger and better known. Perez is more experienced and lauded for his effectiveness.

“We’re vetting VP candidates!” said Arturo Vargas, the head of the National Association of Latino Elected and Appointed Officials, at its recent annual convention in Washington.

Perez and Castro both spoke to the influential audience of officials from every state. So did another high-ranking Latino, Rep. Xavier Becerra (D-Calif.). There was a lot of talk about the record 27 million Latinos who are eligible to cast a ballot in November and efforts to turn out that vote. Many spoke about how much they hoped that 2016 is the year that, for the first time in history, a Latino is named a vice-presidential candidate.

In his address, Perez slams the recent Supreme Court deadlock that effectively blocked Obama’s plan to shield millions of undocumented people from deportation.

“It frankly breaks my heart,” Perez tells the crowd. He says the court’s decision strengthened his resolve to fight for more “fairness” and an overhaul of immigration laws. He said comprehensive immigration reform “is a ‘when’ question in this country. It is not an ‘if’ question.”

“He’s motivational and strong against racial profiling,” said Jose Yepez, a city council member in Somerton, Ariz., who had never heard Perez speak before. In a year when Trump is talking about deporting millions of undocumented immigrants, Yepez said Perez was right to urge Latino leaders “to stand up and be active.”

Less than 24 hours later, Perez and Castro are together again, this time at the Center for American Progress, the think tank founded by John Podesta, the chairman of Clinton’s campaign.

They are talking to a small crowd about ways to help recently released prisoners find jobs and housing. Attorney General Loretta E. Lynch sits between the two Cabinet secretaries.

“I am ambivalent about using the term ‘second chance’ for prisoners, because many of them never had a first chance,” Perez says.

The whispering in the room has nothing to do with today’s topic but instead focuses on whether Perez or Castro will be vice president. There is a lot of chatter about Perez, who is the most animated on stage this morning, saying “Zip codes shouldn’t determine destiny” and “Just because I pick up trash doesn’t mean you can treat me like garbage.”

“Will he be the next attorney general?” a Democratic staffer in the room wondered. Perez has also been mentioned as a potential Maryland gubernatorial candidate in 2018.

One of the jobs Perez talks about most is that he collected trash to help pay for his undergraduate degree at Brown University. His father’s sudden death from a heart attack when he was 12 drives him “to have purpose,” and he finds that in government work, he said.

His Catholic faith is important to him, and he said he often thinks of what his parents told him: “In order to get to heaven, you have to have letters of reference from poor people.”

Asked what might surprise people about him, he says he still meets up every year with his buddies from eighth grade in Buffalo — as many Republicans as Democrats. He has three children, and his wife, Ann Marie Staudenmaier, is a lawyer with the Washington Legal Clinic for the Homeless.

Pressed again about all the vice-president chatter, Perez just starts talking about baseball. In his parents’ Dominican Republic, “baseball and politics are the national sport,” he says. “So if the commissioner’s job opens up, let me know.”