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I had a similar problem with my '51 a few years back. After a lot of sleuthing, I tracked it down to a slight leak around the glass sediment bowl on the fuel pump. Over a period of time, the fuel in the carburetor would evaporate enough for it to be a little low. While the car ran fine the rest of the time, this small air leak compromised the function of the pump enough to make starting very difficult. I solved it by replacing the gasket on the fuel bowl. I eventually ended up re-surfacing the top of the glass bowl using my surface plate and some "wet-or-dry" sandpaper. It may seem counter-intuitive, but I have found cork sediment bowl gaskets seem to do a better job of sealing than the neoprene ones. I mention this because you mentioned that you had recently replaced your fuel pump and also to show that the strangest things can cause problems.
You can ignore this entire post if you do not have a sediment bowl on your fuel pump
