Here's a bit like I use. I drill the hole in the corner next to the stud, being as careful as possible not to let it break through any more than needed, to lessen the chance of hitting a wire. There are thin fiberglass "pulling rods" sold in different lengths, with a small eye on one end. By drilling the hole next to the stud, the pulling rod will run along the stud, bypassing insulation bats if they're in there. I use the rod to pull a line back into the box, and then the line to pull the wire, with the attachment streamlined as much as possible onto the wire with 33+, so it will ride over any bump.
Once the wire is in the box, I seal around it, in the new hole, with fire caulking.
I like to redo all the wiring in the box rather than just add the new one somewhere. If the device looks like its a couple of decades old, it a good time to replace that too.
https://www.homedepot.com/p/DEWALT-1...14-G/202579649