@nessness -- FWIW, when we lived in the city we had a combined coop/run with about 50 s.f. for 4 chickens, and we let them out when the weather was good. We still had some aggressive behavior, especially in bad weather when they were cooped up for long stretches of time. We installed multiple branches and perches out in the run so at the least the chicken who was trying to escape could flutter up off the ground to a perch.
@Buntastic can chime in with his set up, but 40 s.f. for 9 sounds a bit snug for new hens to be integrating. They may sort it out, but there may be plenty of harsh words and fighting until they do.
In the coop is there anywhere for the lower ranking hens to go to get away? Something to hide behind, or get up on top of? That could help reduce stress and aggressive behavior. I think you probably hit the nail on the head when you said this is the first group of new chickens you've introduced since your flock was young. IME chickens have a
very strong sense of 'family' and 'home'. When chickens grow up together, they're all in the same family -- so 9 may happily share 40 s.f. without much trouble. Chickens introduced later are strangers to them who suddenly appear in their home. When that happens there's a fair amount of unpleasantness until the existing hens finally accept that the newcomers are not going away, and they sort out where the newcomers fit into the flock.
Imagine if suddenly a strange human showed up inside your house, and didn't leave. It'd be very stressful for everyone. Also, the way we'd respond would depend a lot on who the intruder was (child, man, woman), how much space there is, and our personalities. Chickens are the same -- they're individuals with different personalities, some 'spicy' and some mild. :)
I've done many integrations over the years. We had some high drama this spring when we tried to bring a young rooster into our flock of 12 hens. We kept him crated within their sight for a month before we finally let him free range with them. The hens' reactions ran the entire gamut from immediate welcoming and flirting, to aggressive hostility. We let the drama play out for two more months. At the end of that time 10 of the 12 hens had accepted the roo and were mating with him. Two hens -- our alpha and her sister -- were still in rebellion, giving him the middle finger. The rooster beat on those two whenever he could catch them, and when the conflict reached the point of serious wounds, it was a choice for us between him and them. We culled the rooster.
Tl;dr -- I'd try first giving the newcomers somewhere to hide, and if that isn't possible you may need to think about culling the aggressive hen if the behavior is bad enough (she's causing actual injuries, and it's not decreasing). Unless you want to permanently set up another coop? Good luck!