If the problem is the same whether the engine's cold or hot, I think it's way more likely to be in the accelerator pump adjustment. In the couple of seconds accelerating away from a stop, the carb feeds out of the gas in the float bowl, so the fuel pump isn't an issue. If it's breathing badly (dirty air cleaner etc.) it's just gonna run crappy all the time, not just on acceleration. It could also be vacuum advance, but that would generally show up if you were having problems when flooring it, or as a stumble in acceleration after the RPMs come up and the tranny shifts, not so much in mild acceleration from a stop where engine vacuum doesn't come into play much. If you don't fiddle the gas pedal and push it just a bit further down does the engine spit out the tailpipe? If so, it's almost surely not enough fuel on acceleration.
If it's a stock Ford carb, there's two key adjustments that vary carb-by-carb: the accelerator pump lever adjustment, and the accelerator pump stroke adjustment. The lever adjustment is done with the throttle held wide open (engine off) and measuring the gap between the lever and the adjusting *****. Don't know the exact gap for the Ford carb (for a Holley it's .015). The stroke adjustment is all about making sure that the cam that pushes on the accelerator pump is indexed to the same hole as it was before the rebuild ... or you're in for simple trial and error with different settings until the car runs right.