I've tried to remove a faulty 6.5mm headphone jack from a FX-Audio DAC-X6 and ended up removing the solder pads from the board. I'm trying to save this instead of trashing it and buying a new one for $50+. If I order a new headphone socket where can I attach the pins (Right, Left, Ground)? I don't understand what the 4th pin is and where it should be attached.
Follow the traces from each damaged pad and connect a wire between that pin and the other end of the trace.
If it's a TRS jack (and not TRRS), the extra pin may just be for mounting (mechanical) and not need to connect anywhere electrically. It could also be a switched contact. You can figure that out with a continuity meter.
Be aware that this PCB likely has a ground plane which means that any pad that's supposed to connect to ground may not have a visible trace. Check continuity between those pads and a point you know is grounded to be sure.
Next time, be more careful about desoldering before you pull the component out. A desoldering iron with a vacuum pump will come in handy for that type of work.
And when you order the new part, make sure the footprint and pinout matches perfectly. Check the measurements on the datasheet to be sure it will fit properly.