Assuming you still have the starter OUT of the car and on the bench...
Place the starter TIGHTLY IN THE BENCH VISE...(They can throw a wallop when they spin up!
Remove the solenoid..two mount screws and the bolt on the main starter winding screw.
Ground the vice<----READ with a jumper cable, CAREFULLY touch the positive cable to the main winding copper stub coming out of the starter body..The motor should spin up Real fast.........not first slowly and then build speed..(If that happens , rebuild or replace the starter) If it spins right up it's just fine.
Next REMOVE the starter from the vise, and replace it with the solenoid..CAREFULLY hook up your positive jumper cable to the TOP main bolt..Then jumper a smaller wire to the "S" terminal ...
Get your Volt ohm meter, and measure the voltage at the bottom main bolt It should be 12 volts, and go away when you remove the "S" jumper..If not, the solenoid is bad...Replace the solenoid.
If both work well, reattach them and test the whole starter in the vise, If it works fine in the vise, you have a vehicle wiring problem...If it fails in the vise after reattaching the solenoid, the copper stub going into the starter has probably melted the solder under heat and is intermittent on just how much torque is put on the bolt...repair or replace the starter motor.