Vivek Ramaswamy tore across eastern Iowa on Tuesday at the breakneck pace that has come to define his long-shot presidential campaign.

He stopped just long enough at most of the six restaurants and bars on his itinerary to remind voters he’s still in the race, lingering longer at his final stop of the day. He drew praise for his straightforward, bombastic style. And he made humorous quips, promising to finish Donald J. Trump’s mission of draining the bureaucratic swamp in Washington by “bringing the pesticide” to anything that crawls out.

But the day mostly served as a stark reminder of how deeply Mr. Ramaswamy remains mired in a kind of swamp of his own, trailing far behind his rivals for the Republican nomination and stuck in fourth place in most state polls. In Dubuque, a few minutes before Mr. Ramaswamy arrived at a cozy cocktail bar where he was scheduled to speak, one of his campaign’s surrogates asked the 50 attendees how many planned to caucus for him. Only about five raised their hands.

Some voters at his six Iowa events Tuesday wondered aloud whether he was simply burnishing his credentials for a 2028 presidential run or for a position in Mr. Trump’s cabinet if the former president were to win back the White House.

“I think he’s got a really good chance of that,” said Matt Casey, 49, of a possible role for Mr. Ramaswamy in a Trump administration. “He could probably be the vice president real easy.”

Mr. Ramaswamy, who has largely financed his presidential bid with the money he earned from his shrewd pitches to investors in his biotechnology business, can probably afford to remain in the contest as long as he desires. And he has maintained that he will outperform expectations and pull off an underdog victory on caucus night on Jan. 15. He has argued that many of his supporters are young people and other first-time caucusgoers not being counted in the polls.

“I think we’re going to deliver a major surprise,” Mr. Ramaswamy told reporters on Tuesday.

His tactic of hewing close to Mr. Trump’s policies and heaping praise on the former president has won him accolades and respect from Iowa Republicans. But with under two weeks until the caucuses, voters’ support for Mr. Trump seems as ironclad as ever, leaving Mr. Ramaswamy simply as the second-favorite for many.

“I’d like to see a Ramaswamy presidency, but I think he’s got a steep hill to climb,” said Jeremy Nelson, 46, who worried that voting for Mr. Ramaswamy instead of Mr. Trump could help Nikki Haley, who is trying to emerge as the main alternative to the former president. “I don’t want a vote for Vivek in the primary to be a vote for Nikki Haley,” he added.

Still, Mr. Ramaswamy’s pointed rhetoric impressed many on Tuesday, and changed at least a few minds. At the dimly lit bar in Dubuque, he eschewed his typical stump speech and launched straight into a question-and-answer session as his wife, Apoorva Ramaswamy, a surgeon and cancer researcher, looked on.

Mr. Ramaswamy painted himself as a more sophisticated version of Mr. Trump, quoting former President John Quincy Adams one moment and telling a voter that Democrats were “selling us the rope today they will use to hang us tomorrow” the next.

He drew applause when he said that unlike Mr. Trump, he would not be led astray by political advisers who stopped the former president from dissolving various federal agencies, ending birthright citizenship or using local law enforcement to aid in the capture of undocumented immigrants.

Sandy Kapparos, 75, said she was “very impressed” with Mr. Ramaswamy’s wide grasp of various issues.

“He brought up everything,” she said. “He just seemed to know so much about all of it. I was leaning toward Nikki Haley, but now I’m not sure.”

Ben Dickinson, a 32-year-old libertarian from Davenport, who visited a Bettendorf event on Tuesday night with his partner and two children, is planning to caucus for Mr. Ramaswamy. He said he thought the candidate had set himself up well should something happen with Mr. Trump’s candidacy. “If Trump were to drop out, then Vivek would most likely get a lot of Trump’s followers because he hasn’t said anything negative against Trump.”

Mr. Ramaswamy is hardly the first presidential long-shot candidate who has lingered in a primary far longer than expected, and staying in a race can increase name recognition and pay other dividends. Some also-rans, like former Representative Ron Paul of Texas, built fervent fan bases even as their presidential chances dwindled to near zero.

“I think he’ll get his name out there,” Tom Priebe, 75, said of Mr. Ramaswamy’s goal on caucus night. “I don’t know if he’ll do well this time, but maybe next time.”

As his hopes of winning the nomination have faded, Mr. Ramaswamy has resorted to a host of tactics, some of them signaling desperation. He rented an apartment in Des Moines, campaigned through Thanksgiving and has packed so many events into his schedule that he frequently shows up late. His campaign said on Tuesday that he had become the first candidate in history to complete the so-called Full Grassley — a tour through each of Iowa’s 99 counties, so named for the trip the state’s longtime senator Chuck Grassley takes each year — two different times.

Mr. Ramaswamy has also delved into the fringes of the far right, promoting conspiracy theories such as the “great replacement theory” — the racist idea that Western elites are trying to replace white Americans with minorities. On Tuesday, he trumpeted a new endorsement from Steve King, the former Iowa congressman who was pushed out of office by a primary challenger after his history of racist comments prompted the Republican Party to strip him of his committee assignments in Congress.

On Tuesday morning, Robert Johanningmeier showed up to Mr. Ramaswamy’s event at a bar in Waukon, in northeastern Iowa, with a plan. He had a brown “Vivek 2024” hat cued up in the Amazon cart on his phone. Assuming he liked what he heard, he planned on clicking “buy.”

But after hearing Mr. Ramaswamy speak, Mr. Johanningmeier still wasn’t sold, although he said he was wavering. He decided to keep wearing the same hat he had walked in with — a camouflage “Trump 2024” cap. The Ramaswamy hat, though, stayed in his cart.

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