It's best to get cages that are designed to be stacked, if you just stack them up yourself they'll be unstable and likely to fall. There are some at
http://www.birdscomfort.com/flight_bird_cages.html including stacked cages and side by side double cages.
Another option is to get some adjustable shelves from a hardware store like this:
http://www.homedepot.com/p/HDX-4-Shelf- ... U9Ml1dLlYw and put the cages on the shelves. That would probably be cheaper.
How big is the current cage? Some birds will get along fine in a big cage but will fight if the cage is small.