SS Valiant needs warp drive.
I think if we went by TNG standards, yes. But, TOS it could be argued that they operated by a different set of rules
However, it's more like SS Valiant needs a FTL drive, IMHO. It need not be warp as we know it and there is evidence in TOS to point to Impulse able to do just that. Even at a mighty slow 8c, the SS Valiant could've made the trip from Earth to the edge (going "up" on the galactic plane*) in a leisurely 32 years. There is probably a good reason that there were automated stations provisioned ahead of their trip
*Back then, the thickness of the galaxy was 1/2 of what I believe that has been revised today...