[VetteVet]

In cases where the solenoid doesn't click the remote solenoid will usually fix the problem. The easiest way to check it is to jump he solenoid purple wire to the battery cable connection and if it starts the problem is the solenoid windings. It's possible that the connections to the S terminal are poor which would not allow full voltage to the solenoid. If it starts normally when cold then it's most likely the solenoid windings are heat soaked and have too much resistance to pull the plunger in to make the starter connections.

If the solenoid clicks strongly but the starter won't turn or turns slowly then the remote solenoid won't help. The starter windings are heat soaked or the power cables are not making good connections. Sometimes the battery cables look Ok but can be corroded under the insulation.