1992 Toyota Corolla Won't start after driving

I've owned my 1992 Corolla for 14 years. Within the last four years, it will not start after short trips (5-10 miles) during the summer (when outside temps are around 90 degrees, sometimes won't start without driving) and will do the same in the winter on longer trips. Temp gauge is not showing that the temp is high. I've taken it to two different Toyota dealerships and they cannot locate the problem. They replaced all of the main relays and I've put in two new starters with no positive results. I have noticed that if I lift the hood and remove the main relay cover, the car will usually start within ten to fifteen minutes. The #1 fan relay also gets really hot. I've changed the relay twice with no positive results. I love this car and have spent three times what its worth to find the problem. I'm sure I could get another 100,000 out of it if this problem was corrected.