As a dozen top-tier candidates get ready to take the debate stage Tuesday in Ohio, the handful of major Democratic presidential contenders who got shut out are searching for relevancy amid dwindling hopes that they can find their own “breakout” moment on the sidelines.

Former Rep. Joe Sestak, for one, is trying the stunt of walking across the state of New Hampshire — mimicking his trek across Pennsylvania during his unsuccessful U.S. Senate bid in 2016.

“He can keep his weight down, but it didn’t help him,” said G. Terry Madonna, director of the Franklin and Marshall College poll. “If you’re not on the debate stage, you just got to shake your head and say, unless you’re going to fund your own campaign or come up with a huge amount of money other than the investment of time and energy that they put into it, what’s the whole point of it?”

Unlike some of his other lesser-known rivals, which include former Rep. John Delaney of Maryland, Rep. Tim Ryan of Ohio, Sen. Michael Bennet of Colorado, Montana Gov. Steve Bullock, and author Marianne Williamson, Mr. Sestak has yet to qualify for any of the debates.

But the former Navy admiral, who essentially moved into an Econo Lodge in Iowa for a period of time, says he plans to stick it out at least through the Feb. 3 Iowa caucuses.

“Discouraged? Come on. With any kind of exposure we’re going to do all right,” he recently told the Philadelphia Inquirer.

But Bret Nilles, a top Democratic Party official in Linn County, said candidates like Mr. Sestak and Ms. Williamson who announced they were essentially moving to the state have never really had much of a tangible staff presence in the area.

“They got to start catching some excitement [and] some events up here shortly — otherwise it’s really going to become a really difficult challenge,” Mr. Nilles said.

Mr. Madonna said it probably makes more sense for lesser-known candidates to start punching up at their fellow Democrats, since all the candidates are clearly not fans of President Trump.

“The question is, can they have a breakout moment?” he said. “They have to go after the front-runners in a way that’s credible it can’t be based on an act of desperation.”

Sen. Michael Bennet has trained some of his attacks recently on Sen. Elizabeth Warren, who is jockeying with former Vice President Joseph R. Biden for front-runner status, saying Ms. Warren needs to better explain the costs of her universal Medicare-for-all proposal.

The Colorado Democrat said he hopes to stay in the race at least through the Iowa caucuses and New Hampshire primary, and that his record appeals to Obama-Trump voters who might be leery of the Democratic Party’s lurch to the left.

“If history is any guide, the leading candidates today are not going to be the people that do the best in those two contests,” he said in a recent appearance on CNN.

He tried to downplay the impact of the debates.

“Voters in these early states are just beginning to make up their mind[s], so I had a decision to make, which was whether to spend my resources trying to bribe my way onto the debate stage or to communicate with voters in the early states and that’s what I’m doing instead in Iowa and New Hampshire,” he said.

Mr. Bullock, meanwhile, has tried to emphasize his political successes in a state Mr. Trump easily carried in 2016 and said that candidates can’t play the president’s game and expect to win.

“I don’t think the answer is to out-Trump Trump, right? You wrestle with a pig, you both get muddy and the pig likes it,” Mr. Bullock said at a “Politics and Eggs” event in New Hampshire on Friday. “I think that you call him out — you actually call out the facts. You don’t shy from standing up to him. But you don’t make it all about him, either.”

Still, analysts say it will be a difficult path without the kind of exposure and resources that the top-tier contenders can attract.

“At this point, it’s getting increasingly difficult for anybody to break out,” said Thomas Mills, the publisher of a political website in North Carolina who has also been a Democratic consultant.

He did say that Mr. Bullock and Mr. Bennet are candidates who have resumes and messages that have “broader appeal” than those of some of the front-runners in the race.

“Without some sort of big event, it’s hard to see either one of them getting the traction they need,” he said.

Both Mr. Bennet and Mr. Bullock raised more than $2 million in the third quarter, and Ms. Williamson brought in about $3 million.

But those solid-if-unspectacular totals pale in comparison to the top tier of contenders; Sen. Bernard Sanders and Ms. Warren both brought in approximately $25 million.

Allan Lichtman, an American University history professor, said time is running out for such candidates to make an impact in the race.

“Really the only way you could make a hit like that and vault over 10 other candidates would be [to] have some unique, absolutely burning issue,” Mr. Lichtman said.

He pointed to George McGovern’s latching onto the Vietnam War to surge to the Democratic presidential nomination in 1972 almost out of nowhere.

“[He] was able to inspire a whole army of young people without spending a huge amount of money,” he said. “I don’t see any of the also-ran candidates right now with that kind of burning issue.”

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