How can I safely extend an existing outlet through an empty space?
If there is enough slack in the existing wire then you should pull it out up to the new outlet box.

If that's not possible then you can leave an access panel through which you can reach the old box and pigtail a short run of wire. Junction boxes must all be accessible using a screwdriver.

If you don't want to leave an access panel you can use correctly rated inline splices.