Wait, I thought that even interim appointments serve through the end of the old Congress, unless they resign or retire. A governor can turn out a senator before his term is up? I don't think so.
John Ashcroft of Missouri did this to enable Jean Carnahan to be seated early after the 2000 election. She returned the favor by voting against his confirmation as attorney general.
So the answer is that it can happen, but when it is a Democrat relinquishing their seat to a Republican, it is a safe bet that it won't happen.