I get the feeling Trump is very much PR'ing the SoS pick in the sense of building up to it PR-wise, sending the media in all directions, making it clear that he's running the show, making it exciting. It's possible to do both that and pick the right person.
It's also the final stick in Hillary and Obama's eyes. So I think he'll let this one simmer as long as possible. He already knew who it would be weeks ago and probably before the election.