A cloth diaper cover would probably help, but a cover is only as good as the absorbant diaper underneath! So you may need to add a doubler of some sort. I know they make sposie doublers - a friend of mine has seen them in Publix, and they supposedly look kind of like mama pads, but without the waterproof backing. Also, when I used sposies, I tended to need to size up for nighttime (like put him in a size 4 when he was really a size 3). I actually switched to cloth partly because he was leaking in sposies at night on occasion. So now I use a regular size Chinese prefold and a preemie size Chinese prefold, topped with a side-snapping wool cover. Leaks are gone! My DS isn't able to soak 3 sposies at night though. He was just soaking one and leaking a bit out of that.
For your DS, if you wanted to try a cloth diaper solution, you'd probably have to go with hemp.
I definitely recommend wool as a cover though, regardless of what's underneath. It will help absorb some of the moisture, while preventing the moisture from getting all over everything else outside the cover. Yes, it's handwash only, but I'm the queen of lazy, and I find washing wool very easy to do. It just takes a few minutes to wash and lanolize, and I only do it about once every 2-3 weeks. I use the same cover every night, and a little bit during the day. The handwashing is what kept me from trying wool sooner, but now I'm glad I finally tried it!