In Central NC "winter" only lasts a week to 10 days so I left it up. 😉
They like having a dry outdoor place when it's raining.
The new, open-air coop will be fully roofed.
Can you take the doors off and replace them with a wire wall?
You might put up a shade structure. I have a cheap picnic fly over my coop since I couldn't put it into a shady spot until after the house was placed on the property.