As chief of staff to Gov. Ron DeSantis of Florida, James Uthmeier worked behind the scenes to cement the Republican governor’s national reputation as a culture warrior. Mr. DeSantis rewarded him this year by appointing him attorney general.

It was Mr. Uthmeier who announced last month that the state was opening its own immigration detention center in the Florida Everglades. He made international headlines by referring to the swampy, remote location as “Alligator Alcatraz.”

The detention center’s first weeks have been chaotic, and something of a mystery. But it has amounted to a political coup for Mr. Uthmeier. President Trump visited before it opened and praised the attorney general.

“You do a very good job,” Mr. Trump told him during a visit to the center early this month. Then, referring to Mr. Uthmeier, Mr. Trump remarked to others in attendance, “He’s even a good-looking guy. The guy’s got a future.”

The moment offered Mr. Uthmeier hope that Mr. Trump might consider endorsing him when he runs for a full term as attorney general next year, despite Mr. Uthmeier’s close ties to Mr. DeSantis, who ran against Mr. Trump in last year’s presidential primary. And it raised the profile of Mr. Uthmeier, a 37-year-old lawyer who has never been on a ballot, just as the term-limited Mr. DeSantis’s power wanes.

Florida Republicans have speculated that primary races in next year’s midterms could pit a slate of Trump-backed candidates against a slate backed by Mr. DeSantis.