WASHINGTON—Reporters had been shooed out of Mike Pompeo’s meet-and-greet with Kansas veterans in Orlando, Fla., last week when someone asked the secretary of state if there was any way to bring more military installations to his home state.

Mr. Pompeo answered with “a giant Cheshire grin on his face,” according to a person in the room. “I bet your next senator can take care of that,” he said.

Everyone...