The starter relay could be bad.
The YELLOW wire from the Ignition switch to the starter relay could be bad. This could be in the bulkhead connector plug or at the ignition switch.
This is where a simple 12 volt test light is a good thing to have. It has a clamp on the long wire and a sharp probe tip to check connectors.
I beat the heck out of the relay but that didn't seem to help. I may try replacing it, they're not too expensive.
I guess I could see a bad connection being the culprit but I'm surprised that bumping the solenoid would fix the problem every time.
