If you use Mumbad City Grid to move a piece of ICE from the outermost position to the innermost position, is there a way to move that piece to the outermost position again once the run is completed?

Only through other card effects, like Sunset.

If they run the server multiple times you could use the upgrade to reorder the server to your liking. But there is nothing about the card itself that permits you to reset the order when the run ends. The ice has no "memory" of where it was.