From 4L60e common problems. Since your trans is OK when it's warm, maybe the pump is only partially failing.
21. No movement in any gear: pump failure, or total loss of fluid. R&R, or refill pan and find the leak. If out of fluid, avoid running the engine until the trans is refilled to avoid pump damage. To check for pump failure, check fluid level with the engine off, then start the engine and recheck fluid level. If level does not go down when engine is running, the pump is broken.
R&R = Remove & Repair (or Rebuild) = Transmission must be removed to fix the problem and may have to be rebuilt or replaced.