Take the Florence-Kelvin Highway (well-graded dirt) out of Florence.  After 15.8 miles, take the marked turn north onto Cochran Road.  Drive 13 miles on Cochran Road to the river.  It is rocky, but I made it to the river in a Ford Focus.  A High-Clearance Vehicle would have no problem with the road.  There's camping at the river, but its going to be baking hot any time the river is running.