I don't think lessening distance really has much to do with POV. Each POV has it's own distance, and it's simply how intimate you get that brings readers "closer" to the character. I don't believe there is such a thing as "close" or "distant" third person limited. Third person limited is third person limited. There's only how intimately, or how well, the writer reveals character.
I think there's a difference. You can do a third-person limited that's pretty cool or distant--in which the character is described, his actions told, and some of his thoughts quoted or described.
Then you can do a hot or close third-person in which the narrator is well-nigh invisible.
The close one is a little more showing and less telling. I've seen books in which some chapters are from the main character's POV, and very close, while some are from another character's POV, and a bit more distant.
Oh, gosh, these are pretty bad, but the first is more distant and the second, closer.
1. As he walked along, thinking about Jim, he got angrier and angrier. Would a real friend have acted that way? "No!" he said aloud, and received a startled look from a woman entering a supermarket. He concentrated on his anger, pushing other thoughts aside. What a lousy friend Jim had proved to be. What a betrayal!
2. He walked along, faster and faster. How could Jim have betrayed him so?
I thought you were a friend, Jim, but you weren't, were you? "No!" he said aloud, and received a startled look from a woman entering a supermarket.
But remember when-- his mind started, and he shut it up.
He's just no good. He never was any good, only now, I can see it.