Russ Miles, an independent from Rochester, said he agreed more with Mr. Trump than with other candidates on major issues like immigration, though he was uncomfortable with the candidate’s temperament.

“I think about voting for him, but I wish he’d calm down a bit,” Mr. Miles said.

Many independents like Mr. Trump: In a CNN/WMUR poll this month, 34 percent of undeclared voters who plan to vote in the Republican primary said they supported Mr. Trump, compared with 16 percent for Senator Marco Rubio of Florida and 7 percent each for Jeb Bush, Gov. Chris Christie of New Jersey and Mr. Kasich. Some political analysts attributed Mr. Trump’s high numbers to his well-known name, because these voters tend not to think about politics or their candidate options until closer to the primary.

Others are looking elsewhere. Barbara Novak, who voted for Mr. Obama in 2008, said she would vote Republican this time because she wanted to help Mr. Christie’s chances to be the party’s presidential nominee. Ms. Novak said she was concerned about national security and was not certain she would vote for Mr. Christie in the general election in November, but would like the option.

“He’s very forceful, and by next fall, I think that’s exactly the kind of leader we may need,” said Ms. Novak, of Hampton Beach.

Fifteen of the last 20 winners of New Hampshire primaries went on to become their parties’ nominees. One who did not, Mrs. Clinton, who won the state in 2008 but lost the nomination to Mr. Obama, is hoping to impress independents by focusing keenly on local issues like the heroin epidemic and mental health care needs.

“Independents are hugely important to the primary and they want to be heard, and Hillary has been listening to them closely,” said Jennifer Palmieri, a Clinton spokeswoman.