I went through a bout of that with my last notchback. Tried about 3 starters and the same number of solenoids. The problem turned out to be the solenoid. I kept buying the new solenoids at Autozone and Discount and they would work for about a week and then back to sticking. I finally bought one from Ford and the problems went away. I took the dead ones apart and found that the contacts were welding themselves closed. That's why the starter would run on. Sounds like that might be what yours is doing if it will work again for a while after you tap on it.