Well if you double up to $4 and everyone else at the table still only has $2, then it changes nothing right?
If you double up and you have good players with $4 stacks to your left then consider moving.
If you double up and you have fishes with $4 stacks to your right then consider staying.
Also, the game is very different with 200 or 250bb effective stacks compared to 100bb stacks, in the same way shortstacking is not the same as playing with standard 100bb stacks. So it also depends on your level of comfort playing deep.
There is a post somewhere on here about the considerations when playing super deep, can't find it just now.