The mention of Rodgers and "State Fair" brings to mind the fact that Alfred Newman was the music supervisor and conductor for both the 1945 and 1962 versions.
For some reason, this doesn't happen very much (the only other one I can think of is Goldsmith's Twilight Zone since he did some episodic work for the original series).
And that doesn't really count because none of the episodes that were remade had original scores first time around.
