As indicated, the rod slides into the visor body. Where the rod slides in, it slides into a bushing of sorts. It is a split bushing, probably, which the tension screw, if it has one, tightens against.
In the world of middle-80s GM cars, that bushing is plastic and about 1" long, with a flange that keeps it from being installed farther into the visor body than it should be. On those cars, the "stub" for the sunshade support is only several inches long, rather than longer (as the earlier ones were).
When the support is removed from the visor body, the upholstery typically covers the flange of the bushing, so it's hard to see, but it's there. It can be removed with a pair of needle-nose pliers. The new one will be a little hard to install, just as the visor body on the support will be, too.
CBODY67