By now, many voters have heard that Mitt Romney once put the family dog, Seamus, in a crate and strapped him to the roof of a station wagon. But far fewer have heard that Romney and his sons once raced across a dark, placid lake on Jet Skis, “Baywatch”-style, to rescue strangers and their dog, McKenzie, after their boat capsized.

Or that Romney once temporarily closed the Boston headquarters of his private-equity firm to round up his co-workers, accountants and lawyers and fan out across Manhattan to search for Melissa Gay, his Bain Capital partner’s missing 14-year-old daughter.

Or that as a volunteer lay pastor of his Mormon congregation, Romney spent years counseling neighbors on their marriages and adoptions, helping the unemployed feed their families, and ministering to the sick and the addicted.

The lesser-known stories have surfaced occasionally in profiles of the former Massachusetts governor. But they have not blossomed into any kind of gentler portrait of Romney, who emerged as the presumptive Republican presidential nominee Tuesday after challenger Rick Santorum suspended his bid.

With the campaign’s focus shifting toward independent voters, especially women, a Washington Post-ABC News poll released this week found that President Obama holds a 2-to-1 advantage as the more friendly and likable of the two candidates.

The long and divisive primary campaign has left Romney shackled by a caricature of a stilted, distant multimillionaire — a quandary that increasingly frustrates some of his advisers and even his wife, Ann.

When Ann Romney recently was asked about her husband coming off as stiff, she said, “I guess we better unzip him and let the real Mitt Romney out, because he is not.”

She told a Baltimore radio station: “It is so funny to me that that is the perception out there because he is funny, he is engaging, he is witty. He is always playing jokes. When I met him as a teenager, he was the life of the party.”

That Romney’s softer side has not stuck with voters may be partly his campaign’s fault. Until now, his advisers thought his personal anecdotes got in the way of his economic message — that tales of altruism would appear frivolous amid an anemic economic recovery.

“This is not a ‘Seinfeld’ race,” chief strategist Stuart Stevens said. “This is not a race about nothing.”

“People care about what you’re going to do for them,” he added. “Will you be a strong leader? Will you be someone who is going to help me get a job? Will you be someone who’s going to change the direction of the country? How off-putting is it when you meet someone for the first time and they pull out their family pictures and say, ‘Let me tell you about my trip to the Grand Canyon’? No, you talk about mutual interests.”

Knowing they are working with a private and sometimes uncomfortable man, Romney’s advisers have not tried too hard to shape the public image of his personality. Instead, they have emphasized his managerial competency and economic plans.

But the reality is that, in modern presidential campaigns, voters expect a level of humanity and verve from their candidates. They gravitate toward those who seem relatable, as former Democratic nominees John F. Kerry and Al Gore learned the hard way. In every election since 1992, the more dynamic and down-to-earth candidate has won.

Now, as Romney prepares to face Obama — who with his recent NCAA brackets has been working even harder than usual to be seen as an everyman — he is trying to open up. He plans to weave more personal anecdotes into his speeches, and he and his family are weighing when to sit down for major magazine and network television profiles.

The campaign is beginning to expand the presence of Ann Romney, who offers moving testimony to her husband’s constancy and character. She has narrated a series of videos about their home life featuring family snapshots.

In the latest video, titled “Family” and released Friday, she talks about her husband wrestling and throwing balls with their boys. “Often I had more than five sons. I had six sons,” she said. “He would be as mischievous and as naughty as the other boys.”

So far, however, the candidate has been reluctant to let down his guard. When he visited “The Tonight Show With Jay Leno” recently, Romney began by talking about math. (He noted that he had picked up nine delegates in Guam and 20 in Puerto Rico.) One of his only jokes was about Santorum, a former senator from Pennsylvania, making a good “press secretary.”

At a town hall meeting last month in Ohio, a man told Romney, “I know you have a heart” and practically begged him to show it. But the candidate shared no specific anecdotes.

Romney is habitually cautious, and has said he knows that a single odd remark or misplaced metaphor can go viral and doom his candidacy. “You’re on all the time when you’re running for office,” he told Leno. “Everything you say is being followed by, you know, a small camera of some kind that someone has.”

But Romney’s advisers said there’s another reason he’s loath to tell personal stories. As Stevens put it, he has “a natural aversion to that kind of braggadocio.”

Alex Castellanos, who was Romney’s media strategist in his 2008 presidential campaign but no longer works for him, said: “He’s a private man in a public world. These days, where every intimate thing in your life is somehow exposed on television, it’s like Madonna. If it doesn’t happen in front of people, it didn’t happen. But Mitt Romney is old school.”

Romney’s family and friends say that in private he can be warm, loose and endearingly goofy. Occasionally that side comes through in public, such as when he made light of a “Saturday Night Live” parody of him as being “raw and unleashed.” The next morning, as he passed out sandwiches at a New Hampshire diner, he quipped, “This is me, just raw and unleashed.”

But usually Romney sticks to his script. The few times he has chatted casually with reporters aboard his campaign plane, such as sharing family traditions like meatloaf cakes on his birthday, his aides required that the exchanges be off the record.

On the trail this year, Romney has not spoken of his 2003 rescue of a family from a sinking boat on Lake Winnipesaukee in New Hampshire. It was Fourth of July weekend and Romney and two of his sons, Josh and Craig, were cleaning up the beach when they heard screams coming from the lake.

The Romneys hopped on their Jet Skis and, together with a few other good Samaritans, ferried the six passengers and their dog, who had been vacationing from New Jersey, back to shore. The story made headlines across New England.

As for the exhaustive search operation Romney helmed in 1996 to find his Bain Capital partner’s daughter, he has spoken of it just once this year — and only after an Ohio voter asked him to. The story is perhaps better known among voters, however, because Romney made a television advertisement about it during his 2008 race, and a pro-Romney super PAC, Restore Our Future, has aired a similar ad this year in some states.

As Romney told it, Robert Gay’s daughter had gone to a party in New York City without permission and had not returned home to Connecticut. Immediately, Romney shut down Bain Capital’s Boston headquarters and set up a command center at a New York hotel.

He had clerks at Duane Reade drugstores stuff fliers featuring a photo of the girl into shopping bags, and he and his team fanned out to find her. “There we were, a bunch of folks in suits walking around in the parks of New York and in the streets and showing pictures and saying, when we saw teenagers, ‘Have you seen this girl?’ ” Romney recalled.

After their efforts made the local news, someone called the hotline Romney set up asking for a reward. He hung up, but Romney’s team traced the call, went to his home in New Jersey and found Melissa in the basement.

Both anecdotes could help neutralize the stereotypical image Romney’s political opponents are pushing of him as a heartless, super-rich technocrat.

“I call it the law of the car keys,” Castellanos said. “Before I give you my car keys to take me somewhere, I want to know where you promise to take me — policy — but I also want to know: Can I trust you to take me there? Politics is not just about policy. It is about character and trust. Mitt Romney has told the policy part of the story, but that sense of who he is, and can you trust him to take you there, is important, too.”