Former President Donald J. Trump’s resounding victory in Iowa significantly raises the stakes of next week’s New Hampshire primary for Nikki Haley and the increasingly desperate contingent of Republicans who want to move on from Mr. Trump.

While Iowa was largely a foregone conclusion at the top, with a spirited battle only for second place, a small but ever narrowing path still exists for Ms. Haley, the former governor of South Carolina, to beat Mr. Trump in New Hampshire. It relies heavily on tens of thousands of independent voters expected to participate in the Republican primary.

Ms. Haley, who got a late start in Iowa, has from the beginning banked her campaign on a strong showing in New Hampshire, and has recently been buoyed by an influx of cash from the super PAC supporting her. The demographic makeup of the state is also much more favorable to her than the more rural and conservative Iowa. She has invested significant money and time here — holding 80 events in the state — and has the support of some its top Republicans, including the popular governor, Chris Sununu.

“She’s on the ground, she’s in the diner, she’s doing the town halls,” Mr. Sununu said. “She’s answering anybody’s questions. Trump’s not doing that. You’re lucky to get him to fly in once a week to do a rally and then get the heck out of there.”