A neighbor has a 1999 Taurus. One day when he ran an errand half a mile away, it would turn over but not start when he tried to come home. A few minutes later it started without trouble.
There was no more trouble until the day his brother drove to buy groceries three miles away. He returned without trouble, but when he tried to run another errand a few minutes later, it turned over but wouldn't start.
The dealer kept it twelve days. It started every time.
Decades ago when a SAAB would turn over but not start, I discovered that the when the ignition switch was turned to START, it switched off the ignition (bad contact). (I remember I didn't have to replace the switch. I may have started by using a jumper wire for the ignition and fixed it with contact cleaner.)
Could that happen with a 1999 Taurus? Perhaps while driving resistance in the switch heats it up so that it may not power the ignition (or fuel system) when turned to START until it has cooled sufficiently.