yes sir you do.
you need a permit on the lower salmon as well (but its a self issue permit)
Any section along the road you can float with a self issue permit as well, as long as you don't spend a night on the river.
If you paddle a trib and intend to paddle out on the MFS you need a MFS permit even if you do not camp on the MFS. If you packraft a tributary and load up your boat and pack it out I think you can sidestep the permit. Or you could float Big Loon and fly out I suppose.
The main salmon is the same except you can float the South Fork and if you do not camp on the main you can just get a SFS permit. It is not lottery of capped but you still need it.
Permits are fairly easy to get later in the season.
Big Creek floaters must obtain a Tributary Permit from the Krassel Ranger District but they are not required to obtain a Middle Fork of the Salmon launch reservation and permit if they will go all the way out in one day (no overnight stays). Contact the Krassel District at 208-634-0600 to obtain a tributary permit.