The first-in-the-nation primary could be the last stand for the anti-Trump Republican.

Since 2016, a shrinking band of Republican strategists, retired lawmakers and donors has tried to oust Donald J. Trump from his commanding position in the party. And again and again, through one Capitol riot, two impeachments, three presidential elections and four criminal indictments, they have failed to gain traction with its voters.

Now, after years of legal, cultural and political crises that upended American norms and expectations, what could be the final battle of the anti-Trump Republicans won’t be waged in Congress or the courts, but in the packed ski lodges and snowy town halls of a state of 1.4 million residents.

Ahead of New Hampshire’s primary on Tuesday, the old guard of the G.O.P. has rallied around Nikki Haley, viewing her bid as its last, best chance to finally pry the former president from atop its party. Anything but a very close finish for her in the state, where moderate, independent voters make up 40 percent of the electorate, would send Mr. Trump on an all-but-unstoppable march to the nomination.

The Trump opposition is outnumbered and underemployed. The former president’s polarizing style and hard-nosed tactics have pushed many Republicans who oppose him into early retirement and humiliating defeats, or out of the party completely. Yet, their long-running war against him has helped to frame the nominating contest around a central, and deeply tribal, litmus test: loyalty to Mr. Trump.

Gordon J. Humphrey, a former New Hampshire senator, was a conservative power broker during the Reagan era but left the party after Mr. Trump won the presidential nomination in 2016. This year, he has produced anti-Trump Facebook videos aimed at encouraging college students and independent voters who, polls show, are more likely to support Ms. Haley over Mr. Trump.

“It’s very big stakes,” Mr. Humphrey, 83, said. “If he wins here, Trump will be unstoppable.”

Campaigning across the state this week for Ms. Haley, Gov. Chris Sununu of New Hampshire, a moderate Republican, argued that the man who remade the party in his image is not its best standard-bearer.

“Trump does not represent the Republican Party,” said Mr. Sununu as he campaigned with Ms. Haley at a rustic event space in Hollis, N.H. “He does not represent the conservative movement. Trump is about Trump.”

Large numbers of Republicans disagree. Mr. Trump, who was trailing in some primary polls only a year ago, now has support from nearly two-thirds of the party, according to an average of national polls by the data-driven news site FiveThirtyEight. In the Iowa caucuses on Monday, Mr. Trump demolished his rivals by nearly 30 percentage points, winning almost every demographic, geographic region and other slice of the electorate.

Elected Republicans have rallied around the former president. On Friday, Senator Tim Scott of South Carolina endorsed Mr. Trump at a rally in Concord, N.H. Even Mr. Sununu — Ms. Haley’s most potent political backer in New Hampshire — has acknowledged that he would support Mr. Trump if he wins the party’s nomination for a third time.

Some of Mr. Trump’s strongest opponents doubt that after so many defeats, they will be successful. Barbara Comstock, a longtime Republican official who was swept out of her suburban Virginia congressional seat in the 2018 midterm backlash to Mr. Trump, said she believed the former president would win the nomination. The only way the party will finally be rid of Mr. Trump, she said, is if he loses in 2024, an outcome she thinks could cost Republicans scores of congressional seats.

“He has to lose and drag down even more people with him on the ballot and that’s the only thing that changes it,” said Ms. Comstock, who opposes Mr. Trump. “You lose, and it’s bad, and you lost for a second time to a really weak guy.”

Recent polling that shows Ms. Haley trailing Mr. Trump by double digits in New Hampshire underscores her uphill battle on Tuesday. Yet even if Ms. Haley can overcome the odds in New Hampshire, she faces the question of what’s next.

A loss next month in a crucial matchup in her home state of South Carolina, where she also trails by double digits, could depress her momentum heading into March, when two-thirds of all Republican primary delegates are up for grabs.

But a victory would give her momentum heading into the Super Tuesday contests on March 5. Twelve of the 16 primaries on Super Tuesday allow independents or other voters to participate, a dynamic that has helped keep Ms. Haley competitive in New Hampshire.

The extraordinary nature of this primary race could alter those calculations. Some strategists say that if Ms. Haley does not win outright, she should hold on until the Supreme Court decides whether Mr. Trump’s name will appear on the ballot in Colorado, Maine and other states. Democrats and some election officials have argued that his role in trying to overthrow the 2020 election should disqualify him for running again.

Still, the strong loyalty Mr. Trump continues to command within his own party has caused Ms. Haley and her backers to make a careful, and somewhat tortured, case for her nomination. Ms. Haley has continued to temper her attacks on Mr. Trump, casting her candidacy less as an existential choice about the future of democracy and more as a moment of generational change.

Speaking to reporters at a diner in Amherst, Ms. Haley cautiously drew a contrast between herself and Mr. Trump. “This is about, do you want more of the same? Or do you want something different?” she said.

New Hampshire primary voters have a history of propelling underdog candidates, including in 2000, when John McCain appealed to independents and defeated George W. Bush, who, like Mr. Trump, was the heavy favorite. A record 322,000 voters are expected to turnout for the Tuesday primary, according to the New Hampshire secretary of state. The surge could portend a spike in participation from independents, who can participate in the primary. So-called “undeclared voters” can take part by choosing a ballot from either party at the polling place.

Part of the problem faced by the anti-Trump wing is one of simple mathematics. A majority of the Republican Party remains staunchly supportive of the former president. But many of the moderate and independent voters who oppose Mr. Trump have voted for Democratic candidates in several election cycles, decreasing the likelihood that they would back another Republican candidate.

These changes have occurred along class lines, with college-educated and higher-income voters largely flocking to the Democratic Party. Mr. Trump’s populist appeals boosted white working-class support for Republicans.

“Many of the college-educated moderates who used to buttress strategies like this for people like McCain in New Hampshire have self-deported from the Republican Party,” said Representative Matt Gaetz, a stalwart Trump backer from Florida. “Like, Nikki Haley Republicans aren’t actually even Republicans anymore.”

In a campaign memo earlier this month, top Trump strategists accused Ms. Haley of creating a campaign “designed to co-opt and take over a G.O.P. nominating contest with non-Republicans and Democrats.”

Mr. Trump has echoed that message as he campaigned across New Hampshire in recent days.

“Nikki Haley is counting on Democrats and liberals to infiltrate your Republican primary,” he said on Wednesday night in Portsmouth. Ms. Haley, he said, is endorsed by “all of the RINOs, globalists, Never Trumpers and Crooked Joe Biden’s biggest donors.”

Ms. Haley has countered that is a lie, noting that Democrats have not been able to change their votes for months and cannot vote in a Republican primary. Any registered Democrat wishing to vote in the Republican primary had to change their party affiliation by Oct. 6. Nearly 4,000 voters did so before the deadline, according to the state’s secretary of state.

But Ms. Haley has also defended her appeal to a broad spectrum of voters.

“What I am doing is telling people what I’m for,” she said during her CNN town hall on Thursday night. “If independents and conservative and moderate Republicans like that, I love that. If conservative Democrats are saying, ‘I want to come back home to the Republican Party,’ because they left it, I want them back.”

At an American Legion hall in Rochester, N.H., several formerly Republican voters who opposed Mr. Trump said they were no longer sure how to describe their political affiliation.

“I am not particularly happy with the way the Republican Party is headed,” said Kristi Carroll, 51, who described herself as a stay-at-home mother and who came to hear Ms. Haley. “I am not sure I am even Republican anymore. I am trying to figure it out.”

Ms. Carroll backed Mr. Trump in 2016 but not in 2020. And she doesn’t plan on supporting him in 2024 — even if the former president wins the party’s nomination.

“After Iowa, I am pretty nervous about the direction of the country, and I am nervous that if Haley doesn’t get the nomination, then I will be voting for a Democrat, which is fine, as long as it is not Trump,” Ms. Carroll said. “Isn’t that awful? I hate to be like that, but that’s the truth.”

A few rows behind her in the crowded room, Chuck Collins, 62, a retired Navy captain and engineer from Alton Bay, N.H., said he used to consider himself a Republican. After voting for Democrats in the last two presidential elections, he now calls himself an independent. Still, he believed a moderate Republican wing would eventually re-emerge.

“We have to have two healthy parties, whether you’re Republican or Democrat,” Mr. Collins said. “You have to have two teams to have a game.”

Michael Gold contributed reporting from Portsmouth, N.H.

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