Staying to rest may trigger several events, based on the characteristics of your followers (kleptomania, insanity, etc.). You may rest several times at the same overhang.
Resting at a Stone Overhang is an inefficient way to gain sanity, as it only brings you 40 and take 5 days of your expedition. Consider using a overhang only when absolutely have to.
One possible encounter is to find a young native with a fever. The native may be treated with either a first aid kit or healing mushroom.