Holley or Autolite or Edelbrock ?
Sounds like sump'thin it holding the needle off the seat.........Trash?!....Need to take it apart for sure.
Is there a steady stream of fuel when you pump the accelerator? If not it could be the accelerator pump is needing a cleaning/rebuild.
WAIT!!!.............Try my shade tree special! Get it running(warmed up too) Using one  hand pull the throttle open and when it starts to rev up......clamp your other hand over the top of the carb. Hold it there until the engine almost dies  and then remove your hand off the top until it revs up pretty good  and repeat a few times.
What happens is...you are creating a tremendous vacuum inside the carb and any trash in the carb will be sucked out(hopefully) If that doesn't work.......may need to look inside.

As I understand what you wrote, the car will only start if you add gas into the carb.  This is for both when the engine is cold or fully warmed up, right?  And since you didn't mention a bad idle, I assume it idles fine after starting.

Well, if it only had issues starting cold then I would say your choke is not working.  And indeed you still need to check that your choke is working and adjusted properly.  However, the choke does not explain the fully warmed engine starting issue.  I would treat these as two different issues and I would solve the fully warmed engine starting issue first.

Actually, I would just assume that the carb needs to be rebuilt.  I would buy a rebuild kit for your model carb, disassemble and carefully clean the carb.  Write down the jet sizes while you have it apart for future reference.  Assume the float level is out of adjustment and carefully adjust it when you reassemble.  Also inspect the choke mechanism for smooth operation while you have it apart.  Then reinstall the carb on the car and see if your starting issue still exists.