Right between the saddle and the long neck on this lamp is the key to keeping it from wobbling — a lock washer. Before that fix, for the longest time, this nice porcelain retro lamp leaned a bit:
There are dozens of posts on the internet that describe how to rewire a lamp, but they often skip the part about including a lock washer on each side of the assembly (inside and outside the lamp) to keep it from unscrewing itself and becoming wobbly over time. I learned this trick from the people at Royal Touch at Coit and Arapaho in Dallas.
Here is a closeup of a lock washer:
Also – when rewiring don’t forget your underwriter’s knot:
Here is the rewired lamp with a new $10 shade from Tuesday Morning: