To me "rewiring the headlight plug" and "adding a wire to the oem plug" are kinda the same... but in any event "adding a wire to the oem plug" is more descriptive of what you actually do.
You add a new terminal (with attached wire) to pin 8 of the oem headlight switch plug, which you then connect to a relay you install... that supplies power thru a fuse you install... to a wire that you run to your new headlights.
Dunno what you'll need to do to run this new wire into the fog lights themselves... it depends on the harness your aftermarket headlights are expecting...but if it has the factory plug-in you'll need to add a new terminal there too. Many after-market kits have a separate wire for the fogs, so no additional terminal required. Mine had a separate wire to the fogs.
It's not plug-and-play: you've got to install a relay, a fuse, additional terminals, and then do a bit of wiring including a run thru the firewall to the front of the car... which is why many people buy a wiring kit.
As per the thread I linked above.