Here's why I did it this way: The 2x3 was stiffer and went up to the roof. It was stronger and easier to handle. It gave me something to attach the hardware cloth to.I put up 2x3 welded wire around my entire run. Then, on the outside of that, I have hardware cloth, to 4 feet.
I can reach my fingers far enough through a 2x3 opening to grab the end of a zip tie. Doing the hardware cloth first would have meant getting someone else to be on the other side, and I was doing this myself.