It was a bit more work than I remembered getting to the rear door handle, but I managed to get some pics before it got dark. Please pardon the dust--dirt infiltration is a real thing!
The problem in my case was a loose hanger holding the cable housing end to the handle. The blue arrows in this photo point to the bracket, which runs behind the white piece:
In stock form, this bracket wiggled around enough that the cable pull from the handle was barely enough to open the door. Interestingly, there was a hole in the middle of the bracket--so I added a screw (green circle below), and this solved the issue:
I typically would have just bought a replacement part, but this was so cheap and easy and has been working fine for years.
So while I don't have any familiarity with the 60 door handle, it seems unlikely that this solution will have an analogue for your issue.
You mentioned that your handle is riveted, but interestingly mine appears to have screws holding it in place from the inside of the door (based on threads I can see from the outside). Perhaps mine has been replaced at some point? If the rivets are anything like those that hold the lock actuators in place, it's really not a big deal to drill them out. The handle should be PN 8C2Z1526604BD, and can be had pretty cheap. Don't forget to buy some rivets too (Ford PN W525172S417, but I'm sure you can use much cheaper generic rivets that are the correct size).