You only check supply status on the "Check Supply" phase. If determined to be out of supply, your unit is out of supply until the next "Check Supply" determination.
So, in your example, if your fighter started out of supply - it is still out of supply even if it flew to a supplied territory on the movement phase. Conversely, your fighter than was in supply is still in supply even if it flies to an out of supply territory.
Same thing goes with armor - if you get an unsupplied armor unit in supply during the movement and combat phase - it still can not move/attack on the mechanized phase.