When I make my next run to the hardware store for more hardware cloth I'll grab some washers and give this a try!
This was my initial plan: staple in place, then attach boards over any seams/staple lines.
Thank you all for the advice! I am doing this big run extension on my own. I'll be honest and say that it's hard for me to be patient sometimes, like when I'm holding up a big roll of hardware cloth and trying to get it to line up neatly, but the roll keeps pulling the whole thing sideways, then holding the roll with one hand while I staple with another all while up on a ladder. I just get really frustrated sometimes, which leads to not moving carefully, which leads to shredded arms and fingers when that roll jumps back at you. (I sort of know the "unroll along the ground" trick, but that's another place where I get impatient and do things the dangerous/stabby way).
I'll definitely try the washer tip and see how it goes!
Question about connecting two sections of hardware cloth (when the edges don't meet at a board): in another thread someone mentioned using hog clips. I'd been weaving wire, like "sewing" the two pieces together. For those who use clips: how far apart do you space the clips?