How did those who found themselves thrusted into the position (via assassination or early/sudden death) do it? I think some just waited until the next election rolled around and selected the VP via normal campaign proccesses. *goes and checks*
Anyways, that sort of musical chairs (like the way Putin did it) would be so obviously corrupt that you'd have to be (metaphorically) blind to not see it. Besides, I don't think Pence enjoys the role of High Priest to Trumpism enough to willingly resign and play it again if he got the top position.
Looking at wikipedia, John Tyler, Milard Filmore, Andrew Johnson, and Chester Arthur (all 19th century) simply left theirs vacant. Teddy Roosevelt left it vacant for the first term (actually the remainder of McKinley's term) but had one for the second who was nominated through the normal electoral proccess. Calvin Coolidge left it vacant for the first term and the VP for the second was nominated via the normal electoral proccess. Harry Truman, LBJ and Gerald Ford followed the same precedent.
I guess Pence might prefer to not leave it vacant, but selecting a VP outside of the normal electoral proccess, even if it required the consent of the Senate, would still smack of circumventing the will of the voters.
edit: Maybe as a compromise, Pence should pick Hillary, hah!