Was the replacement starter a known good one? The symptoms you describe were similar to a starter issue I had. Turned out that it had a bad winding in it--when it was hot, it would not turn over the engine again, until it cooled down. Took me a while to track that down...

But the principle is that when metal heats up, it expands and then continiuty can stop until it cools down and "connects" again. If it were me, I'd check or replace those wires going to the solenoid/battery, including the main battery cables.

Those inexpensive remote starter switches between the battery and the starter solenoid might provide a clue--to some it's a (shudder to think) permanent solution!
Good luck.
-L