Hillary Rodham Clinton greets voters as she campaigns with Sen. Jeanne Shaheen (D-N.H.) at the Farm Bar & Grille in Dover, N.H., on Sunday. (Jim Cole/AP)

The Republican takeover of the Senate could be good news for at least one Democrat: Hillary Rodham Clinton.

Clinton campaigned hard this fall for Democrats, working to boost the party’s effort to preserve its Senate majority — an effort that failed dramatically in Tuesday’s GOP midterm rout.

But many Democratic strategists said the switch to Republican control may have a silver lining for Clinton, helping her better define herself as she shapes a potential 2016 presidential campaign. By providing a convenient foil for her and other Democrats, a GOP-run Congress would make it less imperative for Clinton to highlight her differences with President Obama, these strategists said.

Obama’s damaged, lame-duck condition also makes Clinton the strongest Democrat left standing.

A Republican Senate is likely to “spend a lot of time trying to repeal some of the progress made in the Obama administration,” Democratic strategist Erik Smith said. “That would be a great situation for her, because she could both make the case against the Republicans while currying favor with the Obama base.”

From the Senate to gubernatorial races, here are the key winners in the 2014 midterm elections. (The Washington Post)

But GOP adviser Stuart Stevens, the chief strategist for Mitt Romney’s 2012 presidential campaign, said the notion that an all-Republican Congress is good for Clinton will not bear out.

“I don’t buy it,” he said, because Congress will pass legislation that Obama will then veto, and that will not leave Clinton much running room. “What’s she going to say? ‘I would have vetoed it, too, so I’m going to be the third term of Barack Obama’?”

Two years before the 2016 presidential election, Clinton is in the enviable but precarious position of being the most popular, most famous and most scrutinized contender for a race that everyone assumes she is already running.

The shadow campaign began in earnest Wednesday, when Hollywood mogul Jeffrey Katzenberg’s team began making calls to secure donations for the Priorities USA Action super PAC, which aims to serve as Clinton’s big-money advertising vehicle.

Andy Spahn, a political strategist who advises Katzenberg and other clients, said he has started reviewing donor lists and calling wealthy Democratic backers to get their commitments. “We will be reaching out in the weeks ahead to set up one-on-ones and meet-and-greets to talk about the urgency of the task ahead,” Spahn said.

First, however, Clinton will have to overcome the short-term damage from Tuesday’s Democratic losses. Sen. Rand Paul (R-Ky.), a likely presidential candidate in 2016, said Wednesday on CNN that the election result was “not only a repudiation of the president but, I think, really a repudiation of Hillary Clinton.”

The midterm vote holds lessons for Clinton about which issues most resonate with the grumpy 2014 electorate and which are likely to matter in an election that is still far off, according to political advisers and analysts who are close to the former secretary of state or are watching her. Most agreed that she must fashion a way to run against Washington — a task that will be easier with a GOP Congress.

The GOP gained control of the Senate on Tuesday night, taking hold of the legislative agenda in that chamber. Here are three of the policies Republicans are likely to tackle as they take the reins in January. (Julie Percha/The Washington Post)

The losses also raise doubts about whether the “Obama coalition” of youths and minorities will turn out for anyone but Obama. No candidate, including Clinton, is likely to win as large a share of the black and Hispanic vote as Obama did in 2008 or 2012. But Clinton probably would do better among whites in many states, while possibly expanding Democratic margins among women.

Democrats are hoping the new Republican Senate majority will quickly annoy voters by overreaching or contributing to Washington’s political paralysis. That environment could benefit other potential Democratic 2016 candidates, but perhaps Clinton most of all, strategists and backers said.

“The likelihood that a Republican Congress does either nothing or does the wrong thing I think is an opportunity for her,” said Tracy Sefl, a Democratic campaign veteran who is advising the independent pro-Clinton super PAC Ready for Hillary.

In the week before the election, Clinton stumped for Senate candidates in states including Iowa and New Hampshire, which will hold the first nominating contests in 2016. The Democratic Senate candidate in Iowa lost, but the reelection of Sen. Jeanne Shaheen (D) in New Hampshire was one of the few bright spots for the party Tuesday.

Clinton spent her final day of campaigning with Shaheen in the state, which she won in the 2008 presidential primary after losing badly to Obama in Iowa. Then, as now, Clinton was considered the heir apparent to the Democratic mantle — a whiff of coronation that did not serve her well.

“There’s a lot to be learned from failure. She wasn’t elected, as we all know,” said Madison Waters, 22, who came to a rally in Nashua, N.H., on Sunday to see Clinton. “I think she was great then, but she’s even better now. She’s sharper and more focused.”

As she did throughout her energetic speaking schedule on behalf of Democrats this year, Clinton sprinkled her Nashua stump speech with personal asides and a long view to the future.

“When you look 20, 25 years out and you think — ‘What’s the country going to be like when she’s starting her adult life? What’s the world going to be like?’ — it really does focus your mind on what’s important,” Clinton said, referring to Charlotte, her newborn granddaughter.

She added that she and Bill Clinton “were raised to believe that if you work hard, the American dream was in your reach. You should not have to be the grandchild of a governor, or a senator, or a former secretary of state, or a former president, to believe that the American dream is in your reach.”

Those themes of in-this-together populism and middle-class promise seem sure to be a central part of Hillary Clinton’s platform if she runs. “When she talks about her grandchild, that makes her very personable,” Kevin Smith, 51, said at the Nashua rally. He said he supported Obama in 2008 but is likely to support Clinton now.

The most recent pre-election polling puts Clinton far ahead of potential Republican opponents. The numbers in the latest Washington Post-ABC News survey also show that Clinton remains a polarizing, if nearly universally recognizable, political figure.

When asked whether she would make a good president, 51 percent said yes and 41 percent said no. Just 8 percent said they had no opinion.

Republicans fare less well. For former Florida governor Jeb Bush, the numbers were 26 percent favorable, 51 percent unfavorable and 23 percent with no opinion. For Paul, 21 percent said he would be a good president, 44 percent said he would not and 34 percent had no opinion.

Bill Burton, a former Obama campaign strategist, said that with little room to run as an “outsider,” Clinton probably would tune her message to those of her Republican rivals.

“Her foil has really got to be the Republicans running against her,” he said. “It’s going to be really easy to run against what Rand Paul and [Sen. Marco] Rubio and those other guys are saying.”

Paul is already working hard to contrast himself with Clinton. As the scope of the Republican wave became clear late Tuesday, he posted pictures of her and losing Democratic candidates on Twitter with the hashtag #HillarysLosers.

Also damaged Tuesday was Maryland Gov. Martin O’Malley (D), a long-shot presidential aspirant whose handpicked successor, Lt. Gov. Anthony G. Brown (D), was trounced by Republican businessman Larry Hogan.

Many Democrats want Clinton to put off any head-to-head combat for several months. A few advisers, however, have urged her to defy convention with a fast announcement after the midterm elections.

Clinton appears unhurried. She has said she will decide on a candidacy after Jan. 1.

Democratic strategist Steve Elmendorf, who is not advising Clinton, said she can afford to wait.

“There’s plenty of time,” he said, adding. “The reasons that some people accelerate the timetable is that they want the money” available to official candidates.

Clinton headlined Democratic events that raked in millions of dollars for others this year, and she would be expected to break fundraising records for a general election. Paul Begala, a Clinton White House adviser who remains close to both Clintons, said the couple put “money in the bank” politically with heavy schedules promoting Democrats nationwide this year.

“These two are the most popular Democrats in America, and they put that popularity on the line for folks in trouble in a bad year,” Begala said. “I am quite sure all this campaigning will put a sizable dent in Hillary’s post-State Department stratospheric poll numbers.”

Clinton said almost nothing about her four years as secretary of state while campaigning for Democrats this fall, perhaps in part because it might remind liberal voters of her hawkish foreign-policy leanings.

The post-election season will allow Clinton to address national-security issues more directly and probably to draw sharper contrasts with Obama. Clinton has gone public with her disagreement with Obama over his first-term reluctance to arm Syrian rebels and is expected to air other criticisms if she runs.

That sets up a potential candidacy very much in the centrist Democratic mode that Clinton naturally inhabits, several strategists said — family checkbook issues, job and worker security, women’s pay and health-care equality, plus a muscular projection of American strength abroad.

“The issues terrain in this election looks like it’s going to be a very good fit for a Clinton candidacy,” Burton said. “Given her experience with foreign policy and national security, and the economic issues, I think that she is particularly well suited for this moment.”

But Stevens, the former Romney adviser, cast doubt on that thesis. Democrats, particularly the motivated and more liberal base voters Clinton would need in a primary season, are likely to take a bitter lesson from 2014 that would not benefit her centrist persona, he said.

“I don’t think they’ll look at this and say, ‘These candidates didn’t win because they were too liberal,’ ” Stevens said. “That inherent caution that Hillary Clinton has will be seen as more of the same.”

Jose DelReal in Nashua and Matea Gold and Peyton M. Craighill in Washington contributed to this report.