Unplug the lamp; remove its shade and bulb. Release the harp from its retaining sleeves by squeezing the arms together. Free the socket shell from its base by pressing your thumb hard against the word PRESS on its side; pull off the shell with its insulating sleeve intact. Peel the felt pad (if any) from the lamp base carefully so that you can replace it later.
Using wire cutters, cutoff the existing cord about 1 foot from the bottom of the lamp base. With a knife, strip about 1 inch of the insulation from the old cord still attached to the lamp; do the same to the new cord. Hook the new cord’s wire to that of the old cord; wrap the connection with electrical tape. Pull the socket up to feed the new cord through the lamp.
When the new cord appears, snip off the connection and the old cord. Unscrew the terminals that hold the wires and pull out the old wires. Strip 1/2 inch of insulation from the new cord, then attach its two wires to the socket’s terminals. Slip the metal and insulating shells over the socket base. Knot the cord in the base to prevent strain on the wires. Glue the felt bottom in place. Attach a new plug to the cord.