Nikki Haley, the former South Carolina governor and United Nations ambassador, five months into her first run for president, acknowledges the position she is in.

Though she was the first Republican to announce a challenge to former President Donald J. Trump, she hasn’t spent a dime on television ads, is polling well behind Mr. Trump and Gov. Ron DeSantis of Florida and has struggled at times to make a case for her campaign.

But in an interview on Friday, at a picnic table outside a Veterans of Foreign Wars post in the small town of Lancaster, N.H., Ms. Haley downplayed concerns about her standing in the primary. It’s early in the race, she said, and many voters have yet to tune in to the campaigns.

“I look at it like one goal after another; I don’t look at the end,” she said. “I know that by mid-fall, this is going to be totally different. Once you pass Labor Day, the numbers start to shift. And you can look at history for that. That’s not me just hoping, that’s me knowing.”

As she traversed small towns in the mountainous North Country region of New Hampshire last week, she tacitly acknowledged the uphill race, while also telling her story of overcoming long political odds to win South Carolina’s governorship in 2010, making her the first woman to serve as governor of the state and the second governor of Indian descent.

During her appearances, Ms. Haley also mixed in subtle digs at her primary rivals.

“I did not go to an Ivy League school like the fellas that are in this race,” she told voters in a North Conway community center on Thursday. “I went to a public university.” Touting her degree in accounting from Clemson University, she said: “I’m not a lawyer. Accountants are problem solvers.”

Ms. Haley’s most recent swing through New Hampshire, which holds the party’s first primary, was billed by her campaign as a grass-roots-focused trip, and one intended to introduce her to voters in this part of the state as a former state executive with roots in the rural South, rather than an establishment figure with Washington ties.

Frank Murphy, 54, who moved to northern New Hampshire from South Carolina in 2016, knows Ms. Haley as his former governor. When she introduced herself to the voters crowded into the Lancaster V.F.W. post, he raised his hand within the first few minutes of her speech to tell her he was from Charleston.

“I got to see firsthand what she did to help the economy down there,” he said, adding that he was elated to see her running for president. “To come into a small town meeting like this and to speak to people and to get them to engage and to talk and ask questions? That’s what you want from a politician,” he said.

The challenge for Ms. Haley is that her credentials might be more of a liability than an asset in a Republican primary that seems to be geared more toward personality than policy, with much attention concentrated on Mr. Trump’s legal troubles and Mr. DeSantis’s focus on social and cultural issues.

In small events and meet-and-greets, Ms. Haley spoke as much about her family and personal background as she did about the economy and foreign policy.

She complimented the scenery of the North Country, adding that its close-knit communities reminded her of her hometown, Bamberg, S.C. Her upbringing as a member of the only Indian American family in town — “We weren’t white enough to be white, we weren’t Black enough to be Black,” she said — taught her to look hard for the similarities she shared with others.

Speaking to voters at the V.F.W. outpost in Lancaster on Friday, she poked fun at the southern accents she is used to hearing in South Carolina and tested out a New England twang, asking those present if her saying “Lan-cah-stah” made her sound local.

“Somebody said I sounded like I was from Boston,” she acknowledged, to sympathetic laughs.

Ms. Haley has focused intensely on New Hampshire. By the end of this week she will have made 39 stops in the Granite State, far outpacing most of the Republican field. She is one of the few 2024 Republican contenders — along with Vivek Ramaswamy — to visit the counties in the state’s North Country region, which sits less than 200 miles from the Canadian border and has woodsy, winding roads stretching through the White Mountain range.

Her campaign says it is hanging its hopes on a growing network of supporters and volunteers in the far corners of the state, rather than spending money on radio or television ads — a longstanding tradition of glad-handing and retail politicking.

The strategy has yet to generate much momentum. Most polls of the primary in New Hampshire show her in fourth place, behind Mr. Trump, Mr. DeSantis and former Governor Chris Christie of New Jersey, who has also spent a significant amount of time in the state.

Ms. Haley’s supporters have expressed frustration and confusion that their preferred candidate — whose past roles as U.N. ambassador and governor prompted an event moderator to ask a crowd on Thursday to decide by applause which title he should use to introduce her — has barely polled above 4 percent in most national public polls.

“We don’t understand that because she’s doing so well,” said Beverly Schofield, an 84-year-old Republican voter, clad in red, white and blue, who drove from Vermont with her daughter to see Ms. Haley in New Hampshire on Friday. “It’s very impressive that she’s doing as well as she is. But I’d like to see her move up that ladder quickly.”

Ms. Haley’s standing reflects the challenges of campaigning in this particular primary more than it does her political capabilities, her supporters say. The Republican field has ballooned to a dozen candidates, splintering the anti-Trump vote, while his recent and prospective indictments seem to have only put the former president closer to capturing the nomination. Ms. Haley’s supporters are wondering how the campaign intends to turn things around.

“That’s the question I wanted to ask her,” said Ted Kramer, 81, a retired marketing executive who attended Ms. Haley’s town hall in North Conway. “She’s got to get the profile up.”

Ms. Haley pointed to previous Republican front-runners who later fizzled out, such as Senator Ted Cruz of Texas and former Governor Scott Walker of Wisconsin. The race so far has been painted largely as a two-man race between Mr. Trump and Mr. DeSantis, Ms. Haley said, but voters are likely to sour on one.

“I know the reality of how quickly somebody can go up and how quickly they can fall,” she said. “The shiny object today is not the shiny object tomorrow. So it’s about not peaking too soon.”

She added, “I’m very realistic about what the benchmarks are and what we need to overcome.”

Those markers include securing the required number of donors and funds to make the debate stage in August — which she has done. She also said she would continue to focus on Iowa and New Hampshire while building on the base she has in South Carolina, another early state, where she and Senator Tim Scott, who represents the state, are aiming to leverage similar voter bases and donor networks. The two have not spoken since he launched his campaign, she said.

Ms. Haley also admitted to feeling underestimated in the race. She is often included in conversations about vice-presidential contenders, though she has emphatically said she is not eyeing the position. She also said that many, particularly in the news media, failed to recognize “the street cred that I have,” listing political wins and averted crises seen during her tenure as South Carolina governor and as United Nations ambassador. “I mean, these were no small jobs,” she said.

Republicans longing for an alternative to Mr. Trump made up a large portion of the crowds at Ms. Haley’s events, along with moderate Republicans and independent voters. Few who attended Ms. Haley’s events this week said they were fully committed to supporting her, and many said they wanted to test the political waters, a signature of campaigning in New Hampshire, where most primary voters can expect to hear from every candidate in person, usually more than once.

Ms. Haley, eager to sway some of those who were on the fence, made policy points on the stump and condemned Democrats on race, education and inclusion of transgender athletes. She criticized both Democrats and Republicans for the handling of Covid-19 and chastised Congress, asking voters if they could point to anything their representatives in Washington had done for them.

She also drew on her foreign policy background, saying that the biggest threat to the United States is China and repeatedly criticizing the Biden administration on its approach, folding in terse words for Treasury Secretary Janet Yellen, who is visiting the country this week.

Joanne Archambault, an independent voter who lives near North Conway, said she liked Ms. Haley’s message and saw her as an authoritative speaker on policy issues. Still, she said that Ms. Haley’s talk of foreign policy distracted from domestic priorities.

“I think there’s too much focus about overseas stuff, too much talk about the border and about China,” she said. “Let’s talk about the problems we are facing — you know, gun violence, abortion, let’s talk about those things. Let’s focus on this country and not what other countries are doing.”

Her closing message to voters has been an entreaty to them to tell others to support her. That was good news to Mr. Murphy, the South Carolina transplant who said he was committed to voting for Ms. Haley in the primary in January.

“She said tell 10 people. I’ll probably tell 20,” he said.