According to several of the bed supplier sites, the wood and bolting strips are the same for 60-66. The bed strip bolt kits fit 58-72, and the bed to frame bolts fit 55-72. The frames are different from 60-62 and 63-66. The earlier version is much stronger like a convertible and has torsion bar front suspension. The later frame is a typical ladder style with a conventional coil spring front suspension. For availability of aftermarket parts the 63-66 frame is more desirable, since almost nobody makes anything for the 60-62 chassis, although a later 67-82 front suspension swap is relatively easy.
Regarding the bed swap it should be an easy swap. The only question might be whether the bed mounting bolts line up in the same exact place between the two frame styles. Good luck.
