Turn key, hold and wait.

2001 Chevy S10, 4 cylinder, automatic, 2WD with 60,000 miles. Fires up every morning with no problem, but when you take a short drive and try to restart it, nothing. Nothing happens when you turn the key. The indicator lights go off like they are supposed to, but nothing happens, no click, nothing. My local mechanic couldn't figure it out. It started every time for him, but he didn't drive it, turn it off and try to restart it. His son discovered that if you just hold the key in start position, it always starts in under 30 seconds.
Does anyone have a clue as to what might be causing this?