I have all of my cages wrapped with either a shower curtain or plastic painters sheets from the hardware store. Also keep in mind that the humidity does not need to be, nor do you want it to be, high at all times. It should rise and spike high right after you mist and/or turn the fogger on. You then want it to slowly decrease over the next few hours (as your cage dries out) and then spike again after the next misting.
Keeping the humidity high at all times can actually cause some serious health issues for veiled and panther chameleons, just as not letting the cage dry completely can. They can handle a wide range of humidity, so, IMO, try to ensure that it goes up after misting and takes a little time to come back down and leave it at that. You will drive yourself crazy chasing humidity constantly if you have absolute numbers that you are trying to achieve.