98 Lumina Starting Problems

Car starts fine when cold and after long trips. The problem is no response after shutting down after short warm-ups and short runs to the store. By no response I mean there is not even a click, although all dash warning lights come on when the key is turned. After a cool down period it starts again. 3.1L and auto trans. Does this sound like a sensor problem?

My 96 3.1L did EXACTLY the same thing. I never did figure it out, and since I live in Maine, it got cold before I really had to worry about it. I suspected a corroded wire to the starter, but may have to find out this summer...
If you figure it out, please post and let us know.