Back to the OP request, for suggestions: (and opinions)
The carb being rebuilt can mean, it is cleaned, seals and gaskets replaced and adjustments approximated. It may not mean it is perfectly adjusted on the bench, nor ideally adjusted to your engine, throttle pressure setting, idle setting, choke adjustment, vacuum, and so forth.
You may need to fine tune a few things to get it working well. A well tuned carb (for the engine) runs pretty darn well and seems to just barely under-perform fuel-injected setups. It is rare a carburetor drops in, bolts in to place, and runs perfectly without some minor tweaking.
You're fortunate in that the Carter 4737 is among the least emission finicky.
Unless you're uncomfortable doing so, I suggest rechecking all (or as many as you can) the adjustments of the carb using the factory service manual and rebuild kit instructions (presuming they were furnished by the rebuilder.) A rebuilder can only approximate adjustments without testing it on your actual engine.
Checking these adjustments may be a learning curve for some, but they teach you what to look for, and to listen for. With some patience, I've been blessed to get things running really well following a given rebuild.
Contrary to some other experience shared, I've found the leather accelerator plungers are the best performing when they aren't worn, and properly saturated with fuel. Do check the plunger in the cylinder. It is possible (but unlikely unless the rebuilder goofed) it is the wrong diameter. If it seems tight, and squirts when you move it up and down, it is probably correct. There are only a few sizes, and they vary quite a bit. The wrong one may not move or squirt at all.
Changing to a different carb, jet size, and some of the other suggestions, I would do only when I've given up making an OEM stock setup function correctly.
A vacuum gauge you can read while driving can be really informative.
As many readers may know, when you whomp on the throttle, the secondaries open, and engine vacuum can drop. It is the accelerator pump's job to squirt more fuel in to the motor during this operation to boost the RPM and offset the loss in vacuum and keep the engine accelerating (and vacuum up) during this operation.
The AFB carburetors have mechanical secondaries, which means, you floor it, they open, no matter what. An AVS carburetor is kinda cool, and I love these on 440 motors. The secondaries on the AVS are driven by vacuum, more or less, and while they may open slightly slower than AFB, they are less prone to a stall during hard acceleration when the accelerator pump is not doing its job.
On the AVS, the secondaries do not open as abruptly as they can on an AFB carb, so it is less likely flooring it can open the secondaries, and drop the vacuum faster than the engine can handle.
If you have a good clean carb, that is causing a hesitation under the right load and hard (or abrupt throttle) accelerating action, I agree with others the accelerator pump function and adjustment are the first places to look.
