They probably won't have a decent answer but you might get lucky, I'd call the county extension agent and see if they can put you in touch with someone that maybe can help. This is dependent on a lot of local conditions such as weather and type of soil, plus the answer will change by season. They may have some suggestions on the cover crops though you might be better off with the native cover as it is already established. Some people post a certain square feet per bird number on this forum, but that is for their specific local conditions. Those numbers can vary tremendously.
In Arkansas I had an area about 45' x 60' inside electric netting for them to forage on. Over winter I might have one rooster and 6 to 8 hens, but in the heat of summer that number could grow to over 50, mostly younger birds reaching butcher age. When the weather broke in spring and green stuff started to sprout I'd leave them locked in a 12' x 32' main run for a week and a half or so to let the green stuff get a start, then turn them loose. The 12 x 32 run stayed bare but the netting area stayed green all summer, unless we had a ridiculous drought. Then I'd have to water.
Chickens prefer eating certain plants over others. Just because it as green doesn't mean they had a lot of good forage. They'd eat the good stuff and let the bad stuff grow. I'd have to mow it three or four times a year to keep the bad stuff from crowding out the good stuff. I left them on the netting area all winter and they did some damage from their scratching and eating dead vegetation. It was pretty bare by spring but grew back pretty well.
They did keep some areas scratched and bare. I had several fruit and nut trees in that netting area. They used the shade from a couple of those as a place to dig holes for dust bathing. The area right next to the coop and main run had some bare spots. Dad totally free ranged a flock when I was a kid. Even with no fences expect some bare spots, especially in shade near the coop.
I see Azygous posted while I was typing. I agree. But instead of how many can I squeeze in a certain area, think more of how many do I want and then figure out how much room you can reasonable give them in the coop, run, and forage area. Err on the side of more room than the minimum for each. Then it is going to be trial and error. Work out the system that works for you.