Vivek Ramaswamy, the only top-polling presidential candidate to hit the campaign trail over Labor Day weekend, is enjoying the attention of his newfound status.

Across five events in New Hampshire on Saturday, part of an 11-stop swing in the Granite State, Mr. Ramaswamy drew hundreds of attendees, often exceeding the number of seats or the space provided at venues from a state fair in Contoocook to a country store in Hooksett.

But the crowds and attention being showered on the 38-year-old political newcomer come with something of a caveat: Many of those showing up at his events and driving his rise in the polls see him as a possible vice president or a great future president — but not necessarily a president yet.

“I have socks older than him,” said Pamela Coffey, 69, who came from Peterborough, N.H., to see the candidate in person.

Mr. Ramaswamy, who entered the race in February with little name recognition and no political experience, has campaigned at a grueling pace in early states and adopted an everywhere-all-the-time media strategy that in recent weeks has propelled him to third place in the race, just behind Gov. Ron DeSantis of Florida.

A combative performance in the first Republican presidential debate last month, in which he was attacked more than any other candidate onstage, put a spotlight on him that translated into heightened attendance at his campaign events. But some voters in New Hampshire said they still had reservations about Mr. Ramaswamy’s youth and inexperience.

Mr. Ramaswamy has used his status as the first millennial to run as a Republican candidate to lament his generation’s being “hungry for a cause” — primarily to older audiences. One of the most reliable applause lines at his New Hampshire events was his controversial proposal to require that high schoolers pass a civics test before they can vote.

Mr. Ramaswamy’s “America First” platform and outsider standing are fashioned after former President Donald J. Trump’s, down to his predisposition toward falsehoods. Like Mr. Trump, for example, Mr. Ramaswamy has expressed disdain for President Volodymyr Zelensky of Ukraine: He scoffed at “Zelenskyism” and called the president the “pied piper of Hamelin in cargo pants” as cows mooed in the background at an event in Dublin, N.H.

Pat Cameron of Goffstown, N.H., said he saw Mr. Ramaswamy as a “great candidate” with “a lot of really good ideas grounded in what this country really believes in.” But he added: “I honestly believe that Trump would be the best. Personally, I would have loved to see President Trump take him as his running mate for vice president.”

And Mr. Trump himself complimented Mr. Ramaswamy last week, spurring questions about whether the Republican presidential front-runner would consider Mr. Ramaswamy to run as No. 2 on his ticket if he wins the nomination.

On Tuesday, the former president told the conservative commentator Glenn Beck that he thought Mr. Ramaswamy was “a very, very intelligent person.”

“He’s got good energy,” Mr. Trump continued. “He could be some form of something.”

But Mr. Ramaswamy, who has said repeatedly that he is not running to be second in command, reiterated that stance on Saturday. “I think President Trump and I share this in common: Neither of us would do well in a No. 2 position,” he said at a town hall in Newport, N.H., just after calling Mr. Trump, as he did in the Republican debate, the “best president of the century.”

Despite Mr. Ramaswamy’s frequent praise for Mr. Trump — and repeated promises to pardon him, if he wins the presidency — he has sought to differentiate himself in subtle ways. While Mr. Trump has continued to invoke the 2020 election and the indictments he faces, Mr. Ramaswamy calls for a forward-thinking vision of the United States as a “nation in our ascent” with revived patriotism under a drastically altered executive branch.

And Mr. Ramaswamy has recently alluded to questions of Mr. Trump’s electability, saying on Saturday that the “America First movement does not belong to one man” and that 2024 “can’t be another 50.1 election.”

“I’m the only candidate in this race who can win in a landslide that reunites this country, that brings young people along,” he said in Dublin.

Nonetheless, many voters who came to hear him speak in New Hampshire uttered his name with that of Mr. Trump, unprompted.

“I like that he’s not like a normal politician,” said Reed Beauchesne, 54, of Concord, N.H. “He reminds me of Trump, in a way. I think he and Trump would be great together, actually.”

And for the voters searching for an alternative to Mr. Trump, not being a “normal politician” can be interpreted as a hindrance.

“He’s got some points that resonate with everybody, so that’s wonderful, but my biggest concern is his lack of experience,” said David Leak, 63, who added that he preferred Mr. DeSantis. “Every politician talks great on the stump, the speeches are well rehearsed, but what do they do after they get in?”