I have spent time on several sections (or "passages") of the Palmetto Trail. (Nice trail! not a lot of "stunning views" or anything, but very pleasant).
In my experience relying on shuttles/ride shares/etc probably wouldn't work, unless either you are hiking a section near one of the larger cities (e.g. Columbia or Spartanburg) OR you're hiking the northernmost section where you might be able to take advantage of the Foothills Trail shuttles, which seems to overlap somewhat.
Unless you can convince someone else to join you in a separate car that you can stage at the end, you might consider using a bike: The way I've done this in the past is:
- put my bike on my car's bike rack
- drive to the "end" of the trail
- get on my bike and use the roads to bike to the "start" of the trail (which hopefully won't take too long b/c bikes are relatively fast)
- leave my bike at the "start" (maybe lock it to a tree or whatever)
- hike from the "start" to my car at the "end"
- drive home via the "start" to pick up my bike.
You have to do some planning in advance to make sure that there is a reasonable bike route on the surrounding roads, but actually I think there are lots of parts of the Palmetto trail where this would work great. As you move southeast in the state, this will get progressively easier as the terrain gets progressively flatter. It can still be kind of tiring though, and it relies on you having a bike and a car bike rack, so maybe not ideal. One little idea is that when you get to the "end" try calling a ride share. If it works, great! Leave your bike behind. If not, well, you've got your bike and you can proceed via my steps above.
Hope that helps!