Can I put a second layer of caulk over newly installed caulk to fill a gap?
Ideally it would be one continuous bead of caulking, however for 3/8" of a gap at the very top, you will probably be OK just adding a bit more caulking to fill it in.  However, if you don't want to have to touch it again for a while--and considering the low cost and relative ease of applying it--I think I would be inclined to remove the existing bead and apply a new one that covers the entire length. That way you don't have to worry about water getting behind it.