I have worked around intermittent starter solenoid wiring problems with a screwdriver across the solenoid and starter hot terminals. Only problem is all the sparks that fly during that operation. Then I bought one of these
remote start switches to help in the ER. It's come in handy for multiple uses including cranking the engine to purge the fuel lines after a secondary filter change.
That and a good DVM can troubleshoot many start and electrical problems. Let us know what it turned out to be.