You should be able to pull the inner cable out of the housing from the speedometer head end. It should have a small retaining washer on the upper end behind the square tip. The cable can be removed and lubed with a small amount of white grease on the lower 3/4 of the cable. The grease will work it's way up the cable. Remove the driven gear from the transmission and check it out.
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