When this happens, use the blunt end of the lug wrench to jump between the lug that goes into the starter motor proper, and the bottom lug of the starter solenoid. Make this a momentary jump. THEN try starting with the key. You are basically turning the starter just a little and it will usually allow it to work for awhile thereafter.

If this works, and I expect it will, it is time to change the solenoid and adjust it with the eccentric so that it will catch every time.

Good luck,