back in 70-72 i had a 66 with a floor shifted 3 speed. an i used to bust inner speedo cables every month or so. the hurst shifter made for a sharp bend in the cable. anyway i used to pull the inner cable out an install a new from the tranny end with a light oil on the cable.
Unless there's a kink, or the sheath is deteriorating, it should be doable.
I was always taught to grease them; but plenty of people think oil is better. Run what you prefer.
But removing the complete cable isn't difficult, either.
The worst part is going back through the firewall with the new sheath -- especially if your inner fenders are installed (but, even then... still not very bad).