Nikki Haley is campaigning at a grueling pace as she fights to stay competitive in the Republican presidential contest, crisscrossing Iowa and New Hampshire to find a clear lane forward in a race dominated by Donald J. Trump and his mountain of legal problems.

So far, that path is elusive.

By many measures, Ms. Haley is running a healthy campaign poised to capitalize on rivals’ mistakes. She has built a robust fund-raising operation and her team has cash to spare: A super PAC backing her this week announced a $13 million advertising effort in Iowa and New Hampshire. And at events, voters often like what she has to say.

“She is not pounding the pulpit,” Eric Ray, 42, a Republican legal defense consultant in Iowa, said after watching her speak at a barbecue restaurant last weekend in Iowa City, adding that she had his vote. “She is not jumping up and down. She is not screaming the word ‘woke.’ She is making reasonable arguments for reasonable people.”

Yet as Ms. Haley tries to occupy a lonely realm between the moderate and far-right wings of her party, her attempts to gain national traction — talking openly about her positions on abortion, taking a hard stance against transgender girls playing in girls’ sports, attacking Vice President Kamala Harris — appear to be falling flat with the Republican base at large.

Polls show Ms. Haley stuck in the low single digits in Iowa and New Hampshire, and trailing both Mr. Trump and Gov. Ron DeSantis of Florida in her home state, South Carolina. Nationally, the first New York Times/Siena College poll of the 2024 campaign showed Mr. Trump carrying the support of 54 percent of likely Republican primary voters. Ms. Haley sat in a distant third, tied at 3 percent with former Vice President Mike Pence and Senator Tim Scott of South Carolina.

Worryingly for Ms. Haley, as Mr. DeSantis’s campaign has stumbled and given his competitors an opening, it has been Mr. Scott, her local Republican rival, who has appeared best positioned to benefit.

“I wouldn’t dismiss her just yet,” said Dante Scala, a professor of political science at the University of New Hampshire. But, he added, “When you are treading water among your own party’s voters — that is a problem.”

Allies of Ms. Haley, 51, the sole Republican woman in the race, argue that she has beaten long odds before, stunning political analysts to win the South Carolina governor’s office by climbing from fourth place in the polls and fund-raising.

Her campaign says it has exceeded its benchmarks: At least 2,000 gathered in Charleston, S.C., for the kickoff of her presidential bid. Ms. Haley has held more events in Iowa and New Hampshire than most of her competitors, and her bid is attracting the interest of a wide mix of donors.

When voters ask about how she can prevail, Ms. Haley points to retail politics — “get used to this face, because I am going to keep on coming back” — and her financial strength. Her top competitors have spent millions of dollars, with little to show for it, she suggests, because few voters have been paying attention in these early summer months.

“We haven’t spent anything,” she said in Iowa City, declaring her campaign was about “to kick into full gear.” She added, “You will see me finish this.”

But Mr. Trump poses a different type of obstacle for her, and for every other Republican candidate playing catch-up.

Ms. Haley, who served as United Nations ambassador under the former president, has carefully calibrated her approach to Mr. Trump and his unwavering followers. Delivering many of the same broadsides he does, but cloaking them in calm tones and plain language, she has alternated between criticism and praise of the former president.

Her unwillingness to directly confront Mr. Trump has drawn criticism from some anti-Trump Republicans. Former Gov. Chris Christie of New Jersey recently compared the reluctance of Ms. Haley and other candidates to mention Mr. Trump to the “Harry Potter” world’s fear of uttering the name “Voldemort.”

“Nikki, it’s OK,” Mr. Christie said. “Say his name. It’s all right.”

Ms. Haley fired right back, saying: “I’m not obsessively anti-Trump like he is. I talk about policies.”

At a gathering with six other Republican rivals on Sunday in Iowa — though not including Mr. Trump — Ms. Haley mentioned the former president in passing, not as a 2024 rival, but to recall how he “lost his mind” in delight over a briefing book she prepared while serving as his U.N. ambassador.

Her speech was heavy on foreign policy, most notably warning that China was outpacing the United States in shipbuilding, hacking American infrastructure and developing “neuro-strike weapons” to “disrupt brain activity, so they can use it against military commanders.”

Ms. Haley has spent years toeing the line between the Reagan-Bush neoconservatism she once sought to emulate and the Trump-centric politics of today’s Republican voters.

During the 2016 election, when Mr. Trump first ran, she did not support him in the Republican primary or his pledge to build a border wall. But she eventually said she would vote for him and later agreed to serve as his ambassador. She left on good terms at the end of 2018, receiving a rare glowing review from Mr. Trump in an administration in which staff turmoil and turnover were rampant.

After the Capitol riot, she faulted the president. But she later contended that he was needed in the Republican Party and lavished praise on his approach to foreign policy, including his dealings with President Vladimir V. Putin of Russia and Kim Jong-un, the leader of North Korea. She has since echoed Mr. Trump’s hard-line immigration message, including an idea to deploy the military against drug cartels in Mexico.

In recent stump speeches and political events, Ms. Haley has turned China — and not Mr. Trump — into her foil, amplifying her attacks on the Biden administration for its attempts to thaw relations with the global superpower.

As governor of South Carolina, she lauded and welcomed Chinese companies, helping them expand or open new operations in the state. But on the 2024 trail, she has argued that this investment accounted for less than 2 percent of the jobs and projects her administration brought in, and that she did not learn how dangerous China was until she became U.N. ambassador.

“I’ve been across the negotiating table from China,” Ms. Haley told an audience of more than 50 people at a manufacturing company in Barrington, N.H., promising to crack down on the “Chinese infiltration at our universities” and the importation of fentanyl from China across the Southwestern border. “They don’t play by the rules, they never have.”

A bright spot for Ms. Haley is her fund-raising. She raised $7.3 million through her presidential campaign and affiliated committees from April through June, according to financial filings that revealed her strong appeal to small donors. Her robust network of bundlers, or supporters who raise money from friends and business associates, includes 125 such backers. Forty percent of them are first-time bundlers, and the group includes powerful women in business and politics, according to her campaign.

Jennifer Ann Nassour, one of her bundlers and a former chairwoman of the Massachusetts Republican Party, said Ms. Haley was in a prime position to break out at the first Republican debate this month.

“No one wants to see another Trump-Biden showdown,” Ms. Nassour said, adding that it was “not good for democracy.”

At the town hall event in Barrington, Toby Clarke, 64, asked Ms. Haley a question weighing on many G.O.P. voters who would like to move on from Mr. Trump: How can the Republican Party come together and avoid splitting its primary results in a way that hands the nomination to the former president?

“Everybody is worried that this is going to turn into 2015 all over again,” Ms. Haley responded, assuring Mr. Clarke that the field of Republican candidates was smaller and that she was meeting the necessary benchmarks to pull ahead. “It’s not going to be 2015 all over again.”

At an event at a vineyard in Hollis, N.H., later that day, with attendees shielded under umbrellas as rain poured from the sky, Ms. Haley expressed optimism, promising to outwork her rivals.

“Republicans have lost the last seven out eight popular votes for president — that is nothing to be proud of,” she said. “We need a new generational leader.”

Trip Gabriel contributed reporting.