You ruled out the switch. So I suggest checking the wiring that goes to the starter, specifically the small red wire at the battery end. That wire tends to corrode and may break. I am just trying cheaper alternatives to replacing the starter motor, which may or not have a solenoid problem.

There are times when the light grease in the solenoid plunger gets dirty and stiff in cold weather, or internal wires get shorted when the temp is really cold. Then it is time for a new/rebuilt starter.
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Keeping it running is better than buying new