I have the same problem with my 66 Ranchero. I put an electric fuel pump in line after the mechanical pump. I ran the wires up to the battery with clips on the ends. When I want to start the engine I clip the positive to the battery run the pump until the bowls are full disconnect the clip and drive away.

The fuel is evaporating out of the bowl via the vent.  A squirt bottle with some gas down the vent hole also works if its been sitting for a white.  Electric pumps are fine, but IMO there's no need to run an electric AND a mechanical pump.  You are adding complexity to the system.  Alternatively just switch to an electric pump.  I never have an issue firing up my tractor after it sits for 5-6 months because its has an electric pump.  Cycle the key once and then fire her up. 

My 69 Mach was fine as long as it was a daily driver. We had a detached garage and when we opened the door there was a strong smell of gasoline. Since the mower was next to the car, I thought that was the source, since there was no evidence of a leak on the car and it always ran well. Eventually we moved to a house with an attached garage and still got the gas smell even though the mower wasn’t in the garage. Years go by and it was retired from daily driving. If it isn’t driven for 4-5 days the smell is gone and it has the same symptoms as yours. If I take the air cleaner off and squirt some gas in the bowl vent, it starts right up. Everything is original, and it has been the most trouble-free car I have ever owned- until we stopped driving it.
I bench tested the carb for several days and it didn’t appear to be leaking inside or out. Someone on VMF said that gas shouldn’t evaporate in 5 days, so I performed a test with a glass container, put gas in it and then a metal plate on top with an appropriately sized hole the equivalent of the vents, and sure enough the gas stayed in there a long time. My best guess is that it leaks into the intake via a casting flaw, or through one of the air bleeds. I have rebuilt that carb many times in 56 years, but I’ll be hanged if I can find it.
I’m going to test my theory by replacing the original 2V carb with a 4 barrel and see what happens. I’ve been assured by several people that it should retain the fuel for quite a while, and not smell, even if not driven.

Sal, I dont remember you spraying it, but I dont think I was in the garage when you started it.  I think I was moving my truck

 

Be sure the vent in the fiel cap is not plugged.  A little bit of vacuum will develop in the gas tank and when cools down it will suck fuel back to the tank. As a test, next tim you park it, take the gas cap off and see if anything changes when you go to start it after sitting.

Other thing I have seen, mainly with Holley, is a seeping accelerator pump diaphragm.  It can leak just enough to evaporate on the intake manifold and you never see anything wet leaking. But it will eventually drain the bowl.

Best thing is to start the car every couple of weeks, rain or shine.

Hey - you could always rig up one of these like is used in small aircraft to start the engine.

https://www.aircraftspruce.com/catalog/appages/primerplung.php
Three or four good squirts below the carb and she lights right off!

https://youtu.be/CWXvvKMzA68?t=4

A good video is worth a thousand words Terry.......N-joy
6sally6