It sounds fuel related. It could be in the tank where junk in the tank plugs the line when it runs for a while but then when it's shut off it settles away from the pickup until. I had a 33 1 1/2 truck do that. Could also be the pump, could be the carb. I found my problem by giving it a different can of fuel to run on, when I didn't have the problem on the gas can I knew the problem was in the tank. Just be really careful, have somebody hold the gas can while you're testing it.
I inherited my dad's '32 roadster butthen I'had a stroke, so I have not done anything with it. Right before the stroke I was going to get it running, it had been at least 15 years since it was last driven. It ran fine when it was parked but now the gas line is plugged solid with rust, I couldn't even blow air down through it, it's plugged up solid. I gave it an auxiliary can of gas and it started right up but now it has one of the valves suck stuck open. That's all from letting it sit and just not starting it every couple of weeks. I have a real 66 GT-350 that I have let sit. Every time I want to drive it around the block I have to drain the fuel, out of the tank unstick the floats and then I then can drive it. I have a dozen neat collector cars, most neat restorations which I could have done but since my stroke I just lay on the couch, so that's my excuse. I was a ASE mechanic for 30-years. Gasoline goes bad in a year or maybe less. Gaskets in carbs dry out, fuel pumps diaphragms dry out and suck air. The best thing you can do is drive them and keep fresh gas in them. Just keep a 1/4 tank in them so you're constantly using it up and putting fresh gas back in them and don't have a stinking stroke! It's kind of like plugging a fuel line in your head, you collapse to the ground unconscious, my daughter found me a couple hours later laying in the driveway and when I woke up in the hospital. Two years later and I'm still not feeling right. At least I got my license back, they took it a way for six-months.
Beautiful Coupe!
