the rotor moves back and forth a good inch.
I hope not. If it really did, that means the distributor shaft is at least half worn away.
Getting a new rotor clip with the tab aimed upward will help to remove some slop between dizzy shaft and rotor (but to the tune of about 1/16 inch or less. An old clip can be repaired and re-stressed.
Notice: The reproduction '54/55/56 distributor caps are made of a substandard material and are a very tight fit on the distributor, but they also tend to SHRINK due to heat and become even tighter so that it is difficult to get the cap aligned with the slot in the top of the dizzy housing.
The repro rotors available for these don't have the rounded edges on the rotor contact tip which swings into contact with the dizzy cap towers like the OEM rotors did, thus they are much less tolerant of slop in the dizzy shaft or rotor. If there is excessive slop, the rotor will break off the dizzy shaft inside the dizzy cap upon attempting to start the engine.
The new rotors that I have received, I have filed down the edge of the rotor tip to match the radius of and old OEM rotor to prevent any problem. It is best to get a short piece of copper tubing and some electroplating acid to re-plate the filed corner of the rotor with copper, or else it will corrode very quickly in areas of high humidity.
If you have way too much slop in the distributor shaft, you will need a new shaft bushing. I think they are still available from repro parts suppliers.