Sporadic starter motor problem after driving 10 miles

Have you checked the sylanoid switch some are located on the starter but most fords are under the hood see if the wires are tight , feel of the switch and see if its getting to hot . it doesn't cost much for that . also it could be the electrical part of the key switch inside the steering wheel column . get a hanes manual or pull it up on YouTube .

Good ideas. I think it is heat related. In the past I did have a problem with the ignition switch. The solenoid may be it too. How about a sensor that cools down after 10 minutes and then allows the system to start?? JH

Even though you got a new starter it sounds to like a BAD starter if it is new it should still be under warranty take it back and get another starter something inside the starter is bad when it gets hot engine tempature it's overheating the starter. I had that did the same thing changed the starter never had problem again it exactly the way you explained the problem.