There is kayak or canoe access about ten minutes up the road at the Alston-Palmetto trailhead. Take a left out of the campground and turn left on the first highway you intersect with, highway 213. On your left is the entrance to the trailhead.
Lake Monticello isn't 25 mins away. I'd give it 15 mins at a cruising speed of 60 mph; at the most. Google says it's 14 miles away to from the camp ground to the center of the lake. If it takes you 25 mins to drive 14 miles, straight up State hwy 215 South, and with a speed limit of 55 mph; in order to get to the lake, you might be going the wrong way.
No access to the river from the campground but broad river access is about 15 mins away and lake Monticello is roughly 25 mins.
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