Could be a number of things, but I've noticed that just about all Chevrolet computers "learn" a short term closed loop fuel map that seems to be overly dependent on (sensitive to) the IAT (inlet air temp), such that when the outside temp changes 20 or so degrees between startups, it can take a while (several warm-up, cool down cycles) for the engine to adjust its fuel delivery map. This map seems to take several trips to adjust, and when the weather fluctuates frequently, as it does in the Spring (and Autumn), this slow learn acts like a dog chasin' its tail, never getting it right.
Consider the issue you describe -- a cold start is fine because the cold start map is probably static and not learned over time -- once warmed up, and parked, the next start will use what is an old, learned warm start map, which indicates cold inlet air temps and a richer fuel map than required for the now warmer inlet air -- while the sputtering indicates not enough fuel, it may also be bogging from a too rich "choke" function. If opening the throttle (giving it gas) while starting can stop the sputtering and keep it running, the extra air is what it needs.
With all EFI, depressing the "gas" pedal is really only giving the engine air, the PCM actually gives the fuel in "response" to the air it detects.
Try getting the short term map to "reset" -- I think you can do this by removing the PCM fuse for about an hour.