I disconnected my battery for 2 weeks while working on car. REconnected and now all I get is a single click from starter relay area when trying to start.
I'd toss a charger on the battery and/or use a jumper battery.
We'll assume you have 12 volts to the control side of the relay (since you get a click).
Double check the battery-lug connections on the solenoid to make sure all terminals have good contact (to ensure the battery cable has a good connection).
You can put the car in neutral and apply the E-brake. Then, with the key in the run position, jump the two large lugs on the solenoid. If it starts now, the solenoid might have taken a doo-doo.
NOTE that jumping the solenoid bypasses the interlock devices (meaning you can have the car lurch, run over your foot, and/or take off down the road, if you have it in gear while doing this).
Again, charge the battery. Voltage and load handling capability/capacity dont relate very well. A static voltage reading of 12.0 volts isnt great (12.6 is charged).
I'd also double check your battery cables, connections and motor grounding pathways. I use jumper cables in parallel with existing cables to find cable issues. WHen the car starts after parallelling a given cable, that cable and its connection is questionable.
Glad to hear you got it. That's kinda weird because that S-terminal wire is just a very low current trigger to energize the solenoid. It should have either worked or not, rather than allowing a clicking but no-starting situation.