1998 Ford Taurus wont start when warmed up.?

I have a 1998 Ford Taurus. Yesterday I went to the store and when I came back out of the store, I went to start the car and got nothing. Not even a click. Everything else would work. (the battery was OK). A jump wouldn't work. Still not even a click. Then after 45 minutes when the tow truck driver got there, he put in the key and it started right up. I drove it home and when I turned it off and tried to start it again, nothing. Not even a click. This morning I went out to start it again and it started right up. This tells me that something isn't working when the car is hot or warmed up but works fine when it is cold. Does anyone have an idea what it could be. Starter or starter relay maybe? One is $100.00 and the other is $12.