PIERRE, S.D. — With Republicans hungry to cultivate their next generation of national leaders, it is not a Capitol Hill comer or a veteran battleground-state politician who is stirring interest by fusing Trumpism with a down-home conservatism spin. It is the first-term governor of South Dakota, Kristi Noem, a rancher who delights in sharing images of herself shooting pheasants and riding horses.

Ms. Noem began drawing wider attention last year for cozying up to President Donald J. Trump — so much so that she inspired suspicion that she was angling to replace Mike Pence on the 2020 ticket — and hosting him at a July 4 Mount Rushmore event where she gave him a model of the monument with his face included. Her defiance of coronavirus restrictions and her eagerness to project a rugged Great Plainswoman image helped her come in second in a 2024 straw poll of far-right conservatives looking for candidates if Mr. Trump doesn’t run again.

But her approach to politics has sometimes made for rocky relations with her base. Late last month, she got herself into a showdown with the Republican-controlled State Legislature over her veto of a bill barring transgender girls from school sports. And as some party leaders were pressing her to resolve that fight, she prompted eye-rolling at home by inserting herself in an unrelated skirmish — over Lil Nas X’s “Satan Shoes.”

“We are in a fight for the soul of our nation,” she wrote on Twitter, picking a fight with the rapper over his endorsement of $1,000 sneakers featuring a pentagram and, ostensibly, a drop of blood.

If Gov. Ron DeSantis of Florida is widely seen as the brash heir apparent to Mr. Trump, and senators like Josh Hawley and Tom Cotton are attempting to put a more ideological frame on Trumpism, Ms. Noem is trying to cement her place as the only female Trump ally echoing the former president’s trigger-the-left approach among the upper tiers of potential 2024 candidates.