You should check out some CCW Reciprocity maps.
USCCA Concealed Carry Reciprocity Map & U.S. Gun Laws | USCCA (usconcealedcarry.com)
What states do you have? If only NJ, you will not be able to legally CCW all the way from NJ to FL. Basically, only NJ, VA, NC, and GA. Without a FL LTC, you will not be able to CCW in FL.
You should really consider getting Utah Non-Resident CCW Permit (gets you 33 states), FL Non-Resident CCW Permit, and PA Non-Resident Permit.
If you have a NJ CCW, you can go to PA and get one fairly quickly depending on the county you go to. Do a search here for the info.
You can also get Utah and Florida Non-Resident permits fairly easily by attending a class at your Local Gun Shop. However it will take several weeks to actually get.
Even with all of those, you will not be able to legally CCW in SC as they only recognize Resident CCW licenses from states that accept theirs. Since NJ does not accept any other state, SC does not accept NJ. And since the other CCW licenses I mentioned are Non-Resident ones, SC will not accept those. This seems to be a violation of the 14th Amendment.
If you have NJ, UT, PA, and FL, here is where you could carry: