Disconnect the linkage and move the arms on the transmission into reverse. If it goes in easily, start the truck and see if it moves in reverse. If that works, it's most likely a worn bushing. You can try jiggling the linkage and then try going to reverse. I remember a '63 we used to have doing the same thing. I'd jiggle the linkage and away I'd go. It was a worn bushing.
Does it have overdrive? You can't put it in reverse with overdrive engaged. This can happen if the PO wired up the linkage and disconnected the governor and added a switch somewhere like the bottom edge of the dash.
For what it is worth I just picked up a 63' and one of my first projects is squaring away the TOTT. I have some photos from a guy named Jay of his tear down I can share. I have some pics of my basic adjustment attempt on my Flickr site. The three links below from another forum site and on a youtube video on a later model.
