Hey,
I was just looking at it and you know I bet 6 small metal stakes (three going down each side secured with metal clamps (they have a bendable clamp kind for plyers) would hold it up nice and not have any wood. You might have to have the people at the hardware store cut the stakes to right length. Good job on doing something yourself!
The stakes I'm talking about are made for fencing in lots. They have small holes going down them for the clamps to be twisted through to secure to the wire. The stakes we have here for our out door fencing are green if that helps?