Although Donald J. Trump has been out of office more than two years, receding as an all-consuming figure to many Americans, to Margot Copeland, a political independent, he looms as overwhelmingly as ever. She would just as urgently oppose Mr. Trump in a 2024 rematch with President Biden as she did the last time.

“I’ll get to the polls and get everybody out to the polls too,” said Ms. Copeland, a 67-year-old retiree who said she was aghast at the possible return to office of the 45th president. “It’s very important that Trump does not get back in.”

At the same time, Andrew Dickey, also a political independent who supported Mr. Biden in 2020, said he was disappointed with the current president’s record, particularly his failure to wipe out student debt. (The Supreme Court is considering Mr. Biden’s debt forgiveness program, but appeared skeptical during a hearing.) Mr. Dickey, a chef, owes $20,000 for his culinary training.

“I think I would possibly vote third party,” Mr. Dickey, 35, said of a Trump-Biden rematch. “There’s been a lot of things said on Biden’s end that haven’t been met. It was the normal smoke screen of the Democrats promising all this stuff, and then nothing.”

In Maricopa County in Arizona, the most crucial county in one of the most important states on the 2024 electoral map, voters like Ms. Copeland and Mr. Dickey illustrate the electoral upside — and potential pitfalls — for Mr. Biden as he begins his bid for a second term, which he announced last week.

The prospect of a Trump-Biden rematch in 2024 is Democrats’ greatest get-out-the-vote advantage. But the yearning by some past Biden voters for an alternative, including a third-party candidate, poses a threat to the president.

In interviews last week with independents who voted for Mr. Biden, most praised his accomplishments and supported his re-election, some enthusiastically.

But there was a share of 2020 Biden voters who were disappointed and looking elsewhere.

“I think we have bigger problems than just Trump being re-elected,” said Richard Mocny, a retiree who switched his registration from Republican to independent after the rise of Mr. Trump, and who voted in 2020 for Mr. Biden. “Polarization in this country is just fierce,” he said. “I believe in looking at some of the new third parties popping up.”

Recently, the group No Labels, which has not disclosed its financial backers, qualified to be on the Arizona ballot, and has raised concerns among some Democrats that it could field a spoiler candidate who would pull votes from Mr. Biden.

Arizona’s independent voters, a sampling of whom were interviewed after having participated in an earlier New York Times/Siena College poll, are sure to be just as essential to Mr. Biden next year as they were in 2020. His 10,500-vote margin in Arizona, less than one percentage point, was his narrowest of any state. The Electoral College map of states likely to be the most contested in 2024 has narrowed to a smaller handful than usual: Arizona, Georgia, Pennsylvania and Wisconsin.

Independent Biden voters in Arizona said that the economy was certainly a concern, including $5 local gasoline prices and in some cases their own stressed finances. But most Biden voters did not blame the president for persistently high inflation, which they said was largely beyond White House control.

Many passionately agreed with Mr. Biden, as he said in his kickoff re-election video, that the Republican Party has been taken over by the far-right, or as Mr. Biden labeled them “MAGA extremists.”

“The entire Republican Party went so far to the right,” said Sheri Schreckengost, 61, a legal assistant and political middle-of-the-roader, who in the past sometimes voted for Republicans. “Donald Trump changed all that for me,” she said. “The way things are now, there’s no way I’d vote for a Republican.”

Mr. Biden’s victory in Arizona was only the second by a Democrat for president since 1948. Maricopa County was the key to his victory. Mr. Biden flipped 60 precincts that had voted for Mr. Trump in 2016. Most of the swing precincts are in suburbs north and southeast of Phoenix, in an arc roughly described by a beltway route known as Loop 101.

Many suburban residents are newcomers to Arizona and they have transformed the former base of Barry Goldwater and John McCain, both Republican presidential nominees, into a purple state. There are the same concerns about Mr. Biden’s age as there are elsewhere in the country.

In Mesa, a suburb with several precincts that Mr. Biden flipped, Maren Hunt, 48, an independent voter who works as a librarian, said of the president, as she entered a Trader Joe’s one evening, “I think he’s done a lot of good, but, you know, how much more does he have left in him?”

Mr. Biden, the oldest person ever to occupy the Oval Office, would be 82 on Inauguration Day of a second term. Still, if it came down to a contest between Mr. Biden and Mr. Trump, who is just four years younger than the president, Ms. Hunt did not hesitate about how she’d vote. “I’ll make sure to mail in my ballot early, very early,” she said.

Similarly, Dlorah Conover, who would prefer a Democratic candidate in the mold of Bernie Sanders — the Vermont progressive, who declined to run again for president in 2024 after two unsuccessful campaigns — said that in a Trump-Biden showdown, it would be no contest.

“This is a despicable human being,” Ms. Conover, 38, who plans to enter community college this month, said of Mr. Trump. “Biden would win hands down with me.”

Mr. Trump has plenty of support in Arizona. A poll of registered voters in the state in April by Public Opinion Strategies found Mr. Biden leading Mr. Trump by only 1 point in a hypothetical matchup.

Despite the former president’s two impeachments, a civil suit accusing him of rape and defamation, and an indictment related to claims he paid hush money to a porn star, Mr. Trump’s core supporters are dug in.

Lately, he has had increased support among Republicans against his chief rival for the nomination, Gov. Ron DeSantis of Florida. In a Trump-Biden rematch, Americans’ entrenched partisanship means that Mr. Trump could gain as much as Mr. Biden from an impulse to rally behind the nominee.

Barry Forbes, 75, an independent who leans Republican, would prefer Mr. DeSantis as the nominee, but he said he would back Mr. Trump, in part because of Mr. Biden’s costly aid to Ukraine in its defense against Russian invaders — “a war we had no business getting involved in,” he said outside the Trader Joe’s.

Much of Mr. Biden’s 2020 pitch to voters was that he would shrink the deep divisions among Americans, which Mr. Trump had expressly exploited for political gain. Voters seem poised to judge him on the progress he has made.

“I think he’s done wonders on bringing our country back together after the number Trump did tearing us apart,” said Jenifer Schuerman, 39, an independent voter and a fifth-generation Arizonan.

Another independent who voted for Mr. Biden, Joel Uliassi, a 22-year-old student at Arizona State University, was less impressed. “Biden ran on the idea he’d heal the divide,” he said. “He was going to bring us back together. From what I’ve seen we’ve gotten more divided and separated.”

Mr. Uliassi, a music student who plays the trumpet, said he became discouraged about Mr. Biden during the chaotic withdrawal from Afghanistan in 2021, which was when approval ratings of the president first dipped below the share of voters who disapproved, a trend that endures.

“I had hoped this election would not be a repeat of the last election, but it looks like it’s ramping up to be that,” Mr. Uliassi said. “If it was another Trump-Biden rematch, I would consider both candidates more this time.”