I am surprised that it has a push socket. It you want to rewire it the job is simple if you only work from the top. Squeeze the socket casing until the top pops off. leave the bottom cup shaped piece in place. Remove the wires from the socket screws. Pull out the old cord from the bottom. You may have to untie a knot in the top. Thread the new wire back in from the bottom and connect it to the socket screws. Replace the cylindrical piece you removed. If you need to replace the socket with a new one, unscrew the bottom piece of the socket shell. There may be a tiny set screw holding it in place. Loosen that first. Screw on the new bottom section of the socket shell. Connect the wires to the new socket and snap in the top cylindrical part of the socket shell. (don't forget the cardboard insulating sleeve unless you want to see if your circuit breaker is working) Never remove the bottom nut from the base of the lamp. The threaded rods usually have a tolerance of a quarter inch and once they become loose it will drive you crazy trying to get the right amount of thread exposed at each end while holding all the components in place.
I have rewired countless lamps as I was in the business for most of my life and that rule applies to all threaded rod style lamps. Table lamps or floor lamps. Do not do it. Your can slide all the parts up and off for cleaning and then slide them back down.
Tip: If you ever need to wire through an arm like a bridge lamp or bankers lamp or chandelier, drop a beaded chain into the pathway. It will find its way to the other side by gravity. Then tape the chain to the wire as narrow as possible and pull the wire gently through. If it comes apart halfway through; try again without throwing the lamp at the wall.
I hope someone finds this info useful.
