The jump is abrupt, yes. I see why it happened, from a franchise perspective (the whole point is to set everybody in position), and I'm consequently willing to let it go, but it does happen awfully fast.

If this were a launch for a "new TOS" series, and I were a writer on the show, I'd totally want to spend the first season tossing in plots about the characters not actually having sufficient experience for their rank. "In this episode, Kirk has to battle aliens while also dealing with the fact that he forgot to check in with the quartermaster about having sufficient food stores . . . .")