They will naturally search for cover if it's wet and damp and perhaps you could just leave the shed door open during the winter so they can go in there? I'm unsure of your arrangement - do they free-range around? If you cover the run will that restrict your access? I have a couple of coops in the garden but due to a recent avian-flu scare in this area had to enclose my girls in a woodshed but put a run on the side of it. Thinking about it, that seems to be the best arrangement - a summer coop in the garden and they come and go and their winter quarters in the shed where they have more indoor room .
my chooks: Sage&Onion, Roger the randy cockerel, Squawkbox, Bonnie 'N' Clyde, Pancakes, Custard, Omelette & EggNog. Plus four garden cats and another half dozen feral ones that I feed. And a nursery of pipistrelle bats in the verandah roof ..