Sounds like you're well familiar with vapor-lock - I wasn't sure. One of the things that caused a similar issue on my '51 was that mexico-made fuel pump was leaky on the bottom side. The car would leave puddles under it but I could never find the problem. When Weas came to look at it, I started it up and let it run. We looked under the hood and the fuel pump looked like a shower head it was spewing so much fuel! I could never find the problem because when the engine was off, there was nothing leaking. The castings were not true so the surfaces did not line up and it spewed fuel and sucked air. The car acted like vapor-lock, even on 65° days. Once we replaced that pump, the trouble was gone!
Doesn't quite sound like the same symptom though. Since it's fuel related, it may be sludge or build-up in the gas tank and clogging the line. Car sits, fuel recedes/sloshes around, clog goes away. Anyway, keep us informed.