This will show you how easy it is to rewire a lamp yourself instead of paying some place to do it!
Most older lamps do not have a small plug in the base to secure the wire against being accidentally pulled out and risking the chance of a short. For that reason, I hope you wont mind if I suggest a different safety measure to take its place.
In order for the wiring to be UL approved, it requires a UL knot. The knot would be in the wire, within the socket, and just before the ends which will be screwed into the socket. I went to my junk shed and found a lamp to use as an illustration.
It may take a little work to get the socket seated with this knot in place. It will be a tight fit, but it can be done. I know, I have wired thousands of lamps.
Here is a video showing how to tie a UL knot: