The tune and overall condition is important. Bad tune and/or bad condition equals difficult starts, among other things. I don't even use the choke in the winter, even when it gets into the 20s. Two pumps of the gas and it fires right up.
I wouldn't expect a problem with the fuel supply. The carburetor bowl is clearly getting fuel. Otherwise it wouldn't start or run at all. If you're not having issues at cruise, where it will want more fuel, then it just doesn't seem likely. However, an easy way to eliminate the entire fuel system is to use a remote fuel supply, or carefully, with a funnel and length of hose which goes directly to the carburetor. With this you bypass the tank, pump, filter, and all plumbing in between. If it still does not start, you know the fuel supply is not the problem.
I'd verify that the accelerator pump is working sufficiently, that the ignition is working properly, that there are no vacuum leaks, and compression is good. Check the ignition while you are having the problem. It will not help you at all to get it running and then check. An easy way to do that is to have a buddy crank the car while you use a timing light to watch for the flash, which indicates a spark. Or you can pull a plug to look at the color of the spark, which really is best.
So far, the no start problems I've experienced, the ignition was the main cause. There may be other things wrong but usually if fuel gets into the intake and the ignition is working, it's going to do something. It may not roar to life but it will at least sputter or try to start, something.
Another thing, '56 car with a '57 engine, is the distributor from the '56 or '57?
Lawrenceville, GA