Some of the parts will interchange. 67 was a design year change but some of the drive train components remained the same. 6 cylinder trucks used the same short tail shaft 4 bolt side cover trans as the earlier trucks, some V-8 trucks used a 7 bolt side cover trans that was longer. They used different shift rods because of the side cover configuration. Depending on which trans and wheelbase you can also expect some issues with driveshaft length. There is nothing wrong with a 3 speed trans unless a person doesn't like to shift. If the 350 trans works OK I would leave it alone and save a lot of aggravation trying to mix and match parts.
Fred



1956 3100 Pickup/Red/350/3sp OD/PS/Disc Brakes
1957 Bel Air Sport Coupe/Red/355/TH350/PS/Disc Brakes
2017 Silverado LT Single Cab SB/Black/5.3/6 Speed Trans
1947 Willys CJ2A w/F-Head engine