I can't afford to cover my chicken run with a hard top. So I use the bird netting attached to my 2X4 fencing with zip ties. The netting is so light that I don't need much of anything to keep it from sagging. I just run a rope every 10 feet and that is enough support for the bird netting. Except for the winter when wet snow will freeze on the netting and it all comes crashing down. But my chickens don't go outside in the snow filled run during the winter. They stay in their coop.
I have never had any song birds getting stuck in my netting. They will fly in and out of the 2X4 fencing without a problem,
Also, my 2X4 fencing is 6 feet tall, so I don't have to bend over when I go into the chicken run. If you have a shorter fence, or just need more headroom, you could put up a pole in the middle of the netting and have it taper down to the outside fencing. I might try that next year to see if it helps with the freezing rain and snow that crashes my netting in the winter.