yes you can
Except
1. Use duct tape and tape it on securely
2. Ensure that the wire is not broken mid way
3. Ensure that it's free to move and not obstructed in any way
but those 3 things aside........ yeah you can totally pull through a new length while using the old one, that's what guys like me do and hope for
in most cases
I know this is a month old thread, so OP may not be looking anymore -- but in case this is still useful.
This is an extension of point #3 above: The cable/wiring may be stapled to the lumber behind the drywall -- either by design or dictated by building codes. You may want to "peek" up the wall with an endoscope (aka snake camera) to see if you found such staples.
The two photos are on the basement horizontal lumber (easier to photo). First photo is AC power cable, the second one is POT lines (4-wires cat3) from a room 2nd floors up from the basement. During repairs, I saw similar staples for wiring on the vertical 8x4 lumber as well.