Here is something from a very recent VCDL (Virginia Citizens Defense League) news item about parks. Hopefully this should give you the info you're looking for.
Virginia has no restrictions on handgun caliber for defensive arms.. only those used for hunting. And the fact you have a SC carry permit makes you clear for carry here.
Jefferson National Forest is part of the Virginia National Park system and therefore, a state park.
BTW, I visit your beautiful state of South Carolina two to three times a year, along the Grand Strand, and have been doing this for over 40 years. I too, would love to see SC go OC as we have here in Virginia. There are no restrictions here in Virginia as to the number of handguns you may wish to carry. Many people OC their primary gun and CC a BUG. BTW, if you wish to carry into a restaurant which serves alcohol for consumption on the premises, you must open carry your firearm. Concealing into such an establishment will not be allowed until next week on July 1st.
Please enjoy your stay.
2. Status of carry in parks
Bert Nunley asked me to clarify the current state of carry in the following locations. Understand that some of these could change in the near future:
Status of Carry in Parks:
Va. National Parks -- concealed carry with CHP and open carry allowed
Va. National Forests -- concealed carry only with permit
Va. State Forests -- no carry at this time (regulations are in the process of being revamped)
Va. State Parks -- concealed carry only with permit
Va. local and regional parks -- concealed carry with CHP and open carry allowed