“It’s a contest about who can trigger the media and Democrats the most, and Noem is trying to get in that conversation,” said David Kochel, a Republican strategist and a veteran of presidential politics. “It’s, ‘Can I come up with something that’s going to inflame Rachel Maddow and raise awareness among conservatives because Fox will cover how much the left hates me?’” In the post-Trump party, a willingness to confront the news media and do battle with the left, preferably in viral-video snippets, is more compelling to activists than amassing a record of achievement or painstakingly building coalitions. Appearing at the Conservative Political Action Conference in February, Ms. Noem received her loudest applause for saying that Dr. Anthony S. Fauci “is wrong a lot.” Moreover, with Republicans having lost the presidency and both chambers of Congress after struggling among female voters, many in the party want to elevate a woman to their ticket in 2024, when Vice President Kamala Harris is likely to be the Democratic nominee for president or vice president. “She is one of the strongest examples of a great Republican woman,” said Glynis Gilio, a law student, who waited with a few dozen other CPAC attendees for Ms. Noem’s autograph. “We really need that strong female conservatism to pack a punch.”