When engine gets hot, and is turned off, engine will not start until cool. on 1998 Saturn SW2

The car has 166,000 original engine and miles on it. This is the first serious problem I have had with this car. I installed new battery. Check engine light did not come on for problem, so diagnostics will not show anything. Car starts when engine is cool fine. But if I drive for 5-10 min. or more and turn engine off, engine will not start back up until cooled down. What can this be?

Asked by for the 1998 Saturn SW2
I have same problem in a SW2 1997 I can hear the solenoid activate (loud click) but starter will not turn... Indicated Solenoid Contact problem but I can't get at the solenoid. Any help on how to repair contacts in Solenoid will greatly be appreciated.
1 more answer
I had the same problem with my '96 SW2, I tried new battery and that seemed to work for a while but it started again. The problem went away after I replaced the starter. My best guess is that the starter can get weak and the electrical connection are not as good when the starter is hot. Replaced the starter 6 months ago and the problem never came back.