While browsing around the web today, I found some nice screencaps of Bill in an episode of The Fugitive, called Stranger in the Mirror, 1965. The first one is a classic over-dramatic pose (Kirk possessed by Janice Lester, perhaps):
But look at THIS! You can see the EVE under the towel! HOW DID HE GET AWAY WITH THIS??
So the EVE predates Trek. When you book Shatner for a role, you get the whole package, and I do mean package. Wow.
We need to watch this ep.