It maybe the heat from the headers killing the starter solenoid. Check all obvious connections for being loose first before thinking the worst. If this car has been running all along previous without issue it sounds more like a maintanence item.

My car did the same "cold start fine" / "hot start click".

Traced problem back to a burned out idiot light that was not tickling the voltage regulator to activate / charge battery.

My $.02

Starter Heat?
 

I have heard of problems with a starter getting overheated by the exhaust manifold or header and not being able to turn the engine. In most any electrical conductor, increased temperature causes increased resistance. So if the starter gets too hot, the increased internal resistance will make it not pass enough current to turn the engine.

Or, perhaps the heat, now combined with some additional problem (high engine temp., loose connection etc.) may be just enough to cause the insufficient current to start the car when hot. You may want to just do a hand check of the starter for high temp if/when it happens again.

I don't recall hearing where this has been a common problem in Cobras, but in the close confines of some engine bays, it may be necessary to install some sort of shield between the header and starter.

A second thought: Have you checked the timing? When advanced too much, won't this put additional load on the starter? Since you are also having problems with operation at idle after the engine warms up, could it be that something is going on with the timing such that after the engine warms up, the advance function is not allowing the timing to return to normal idle advance, causing the problem to just feed on itself until stopped long enough for everything to cool back down? (I maybe grasping at straws, but just thinking out loud.)

Electric or manual choke? Have you checked the choke adjustment and idle speed screws (if electric)?

... check fuel bowls when it stalls.

Fixed
 

After driving myself silly, I finally have it running. I changed out the coil and the bouncing on the amp meter stopped. However it died on me about a 1/2 mile from my home which naturally was up hill . I would get the same symptoms; turn the key and one click. When the tow truck showed up he put a jumper on the battery and it fired up to get me in the garage. I tried to charge it over night and it would not hold a charge. Bought a new Optima battery, asked Godzilla to get the old one out, adjusted the idle up a few RPM's and took it out today without any problems. I hope that closes this chapter.
I want to thank everyone for their suggestions.

Lou

The bouncing ammeter in the SPF indicates that your voltage system is overcharging which results in MSD ignition issues, killed coils and dead batteries. Sounds like you do not have a 1 wire alternator? If that's the case, the wiring change to correct overcharging ( a simple one, I may add) can be found in the SCOF technical library.
If you own an SPF it is well worth the yearly membership fee to join.
They also have a fix for cooling the carb which may also be part of your heat related issue.

Superformance Owners Association

Buy a new battery..............;)..

I did and that seems to have cured it.

Thanks
Lou