“I think I’m increasingly viewed now as not just a Republican but as something different, kind of a hybrid,” he said in a half-hour interview on Tuesday, shuttling between stops in an S.U.V. “I have people of all shapes, sizes, philosophies and party preferences that approach me. But what does that mean? I don’t know. I’m on television, so all the sudden they want to talk to me. Television moves everybody up, right?”

That remains to be seen, electorally. A University of New Hampshire poll released in February found that 60 percent of the state’s Republicans would support Mr. Trump again in a primary; 18 percent planned to vote for someone else. Another recent poll was more generous to Mr. Kasich, though it still showed Mr. Trump with a clear edge. And while some prominent Republicans in the state, like former Senator John E. Sununu, remain vocal Kasich defenders, few sitting officeholders have made a point of consistently challenging Mr. Trump.

Mr. Kasich’s case against Mr. Trump rests as much on tone as substance. The governor, a longtime supporter of free-trade deals like the Trans-Pacific Partnership, has been sharply critical of the president’s tilt toward protectionism on trade, saying it runs afoul of the Republican Party he knew. He has also lamented the president’s bid to dismantle DACA, the program aimed at protecting young undocumented immigrants from deportation.

On this and other issues, though, Mr. Kasich has sought to contrast himself with Mr. Trump most pointedly as a spokesman for national decency. Rarely was he more animated on Tuesday than during a discussion of the president’s Easter morning tweet declaring, “NO MORE DACA DEAL.” “On Easter,” the governor said on Tuesday, later adding, “Come on, leaders don’t do that.”

Officially, Mr. Kasich says, another run for president is not on his immediate radar, but he is keeping his options open, maintaining the relationships required for any prospective campaign.

Regardless of his eventual choice, Mr. Kasich has succeeded, at least, in installing himself permanently in the grand drama of the Trump era, drawing a pack of reporters to New Hampshire in an off-year on the pretext of a question-and-answer session at a college and ensuring that the news media will continue to track his doings once his term expires in January.