How to move outlet in wall?
That is fine

You can do as you desire -- take the old j-box out and replace it with one facing into the unfinished space.  I'd use a 4" square junction box and mount it so the front lip is flush with the edge of the studs -- this way, if you want to finish the unfinished space later, you can simply slap a mud ring on the box in place of the existing cover, then put a new cover on after the drywall's up.