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

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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?
(2) Answers
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.
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.