WEST DES MOINES, Iowa—Nikki Haley has been waxing sentimental in recent days on the campaign trail about the fun she has had and the people she has met in the past year in this state that hosts the first 2024 Republican presidential caucuses Monday.

Left unsaid is how eager she must be to get out of the state and pivot to New Hampshire, where the nation’s first primary, on Jan. 23, offers her the best prospect of throwing a roadblock in front of former President Donald Trump.