[Jim A]

Lucky you with a new '60 Jimmy. We will want more info about it.
If you have visible shift linkage then you don't have a 4 speed.
That linkage system gives trouble as it wears and happens on many different makes (Ford and Chrysler, e.g.).
Take a close look for play in the various connections. They can be tightened up with home made or OEM bushings.
Good luck.