I'm not sure that warm water increases the humidity. I would increase the number of live plants in the cage. If you can put them in clay pots (as opposed to plastic) it will help. You can even get terracotta pots, soak them in water and put them in the cage without plants. Put one upside down where your dripper hits the bottom to absorb the drip and keep that moisture in the cage a little longer.
You might try covering one or two sides of the cage with plastic. You can cover as many as 3 if necessary.