“We’re running in the Democratic primary and I’m running on the values that make me a Democrat, but there is room for a lot of people,” Buttigieg said. “We’re definitely going to make sure we’re reaching out and cultivating folks who are ready to cross sides.”

Voters not registered with the Democratic Party typically make up more than a third of the Democratic presidential primary electorate in New Hampshire — not the largest chunk of the vote, but more than enough to make a big difference in a tight three- or four-way race, which recent polls have shown.

No candidate has won the state’s primary without also winning a plurality of their party’s voters, said Andy Smith, a pollster at the University of New Hampshire. But “with so many candidates running, the margins for winning become much smaller, so it makes a lot of sense to push for those voters,” Smith said. “Especially if you’re a fresh face.”

“I’m running on the values that make me a Democrat, but there is room for a lot of people. We’re definitely going to make sure we’re reaching out and cultivating folks who are ready to cross sides.”

Back on his bus, Buttigieg said he “often talks about the Obama-Trump-Pence-Buttigieg voter” he won over in Indiana, and though he “makes very specific points about commitments to progressive policy, the fact that people can hear me say that who are considerably more conservative politically and still feel like they might want to support the campaign shows you” what’s “at stake is just how we explain what ought to be done.”

Local officials and activists said they do expect more independents to vote in the New Hampshire Democratic primary in 2020, since the Republican primary isn’t expected to be competitive. That would likely “help any candidates not named Sanders or Warren,” said Kathy Sullivan, a New Hampshire Democratic National Committeewoman. “The big question is: How big of the vote do independents actually turn out to be, and how does it get split up?”

Buttigieg is not the only Democrat making this pitch. Biden premised his campaign on winning back Obama-Trump voters in Midwestern states, and Sen. Amy Klobuchar — who said in an interview with POLITICO that she thinks important to “bring in independents and moderate Republicans” into her party — often points to her track record of big electoral wins driven by bipartisan support in Minnesota.

This week, Sen. Michael Bennet, who received 1 percent in Quinnipiac’s November poll in New Hampshire, said in a statement that he plans “to spend more time in the state than anyone else between now and the primary.”

“Joe Biden, Pete Buttigieg and Amy Klobuchar, even Michael Bennet, are right in line with” what “most independent voters are looking for,” which is “evolution, not revolution,” said Bill Shaheen, a New Hampshire attorney and the husband of Sen. Jeanne Shaheen (D-N.H.), who attended Buttigieg’s speech about veterans in Rochester, N.H. on Monday. “That message absolutely appeals to those voters, and that’s where they have room to grow.”

The competition in New Hampshire for those voters declined, too, when Sen. Kamala Harris cut her staff and shut down her offices in the state earlier this month, redirecting those resources to Iowa.

“I was drawn to Kamala at first, but she doesn’t seem to be going anywhere,” said Sara Groesch, who saw Buttigieg at a canvass kick-off Sunday afternoon in Claremont, N.H. “Pete is very practical, very smart, so I think he can actually get things done.”

Independents are not necessarily moderates, and it’s not just the more moderate candidates who draw from this pool. Sanders and Warren draw their share, while businessman Andrew Yang has said he sees New Hampshire’s open primary as a key piece of his strategy.

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“Since there isn’t a competitive Republican contest, I think [independent] voters may make up a disproportionate and higher than usual vote for Democrats,” said Jim Demers, a Democratic operative in the state who endorsed Sen. Cory Booker. “Targeting moderates and independents could be more fruitful than in previous cycles.”

But Buttigieg’s campaign is arguably among the best positioned to take advantage of that path because he’s sitting on the most resources, raking in more than $50 million into his campaign. This week, Buttigieg kicked off his first statewide TV ad buy in New Hampshire with a pair of ads that lean on his biography and his Medicare-for-All Who Want It proposal.

“We keep sending politicians to Washington to fight for us, but when they get there, they seem more interested in the part about fighting than the part about us,” Buttigieg says in the TV ad, a contrast the mayor has tried to drive against Warren.

The pitch has resonated with Adrian Basora, an independent, undecided voter who attended the mayor’s town Sunday afternoon. To win in 2020, Basora said the Democratic nominee “needs to be able to attract and inspire progressives” and moderates.

Buttigieg could be one of those candidates, Basora said. “It’s not an either-or, and I think a candidate can do both.”