Based on the new info, I'm revising my recommend to join @21hens-incharge . Cover the existing run, call it all coop. Put a new run off of the tall side.
That will provide the most space, use the least materials, and be the fastest fix for lack of space. I'd repurpose the existing chicken wire as roof for the new run, and only minimally structure the run - possibly as simple as 4x4 posts set in the ground, then trimmed to the desired height - probably the existing "high" wall height at the top of the current ridge, so you end up with a cube shaped run (no ducking for you) with a sloped house for snow concerns - and it will shed snow away from the run, also a positive. Finally, will provide some natural shade at least part of the day, from those trees - unless you are already as close to your property line as you can intrude with a structure?
That will provide the most space, use the least materials, and be the fastest fix for lack of space. I'd repurpose the existing chicken wire as roof for the new run, and only minimally structure the run - possibly as simple as 4x4 posts set in the ground, then trimmed to the desired height - probably the existing "high" wall height at the top of the current ridge, so you end up with a cube shaped run (no ducking for you) with a sloped house for snow concerns - and it will shed snow away from the run, also a positive. Finally, will provide some natural shade at least part of the day, from those trees - unless you are already as close to your property line as you can intrude with a structure?