Fighter Umbrella

Escort - If this unit is in the same sector as a friendly Bomber, enemy Fighters get -1 on each attack die when making Antiair attacks against this Bomber

The wording doesn't say the Fighter has to move to another sector immediately. I suppose one could argue that the Fighter doesn't have to use it's ability until the end of the Air Attack phase. But that would be a little messy to allow the First Player to make a move after the Second Player. (Although other SAs do specifically allow that. But they do "specify" it.) Or one could argue it doesn't have to move until at least the end of the First Player's Air Attack step. (I think that would be the best interpretation.) But that would have the same outcome of the fighter no longer being there when the Bombers are attacked during the Second Players Air Attack step.

On the plus side, if we say the answer is "no" and the Escort ability is lost in that sector it would also be true that the escort ability would be gained in the new sector if it also contained some of your bombers.

I think the others are correct that the Escort ability would be lost in that sector if it was your only Fighter with Escort in the sector.