I have raised chickens for 10 years... Just not in a cold climate ... I had a 3 x6 coop in South Carolina and 6 big hens in it and never an issue ever so I know small coops will work for me.
That's one place where weather makes a difference. With your 6 hens in the small coop: were they locked in it all day and all night, or did they go out into a run during the daytime?
During a Michigan winter, there will probably be weeks and maybe even months in which they do not go outside AT ALL. Even if you open the door, they may choose to stay inside rather than walking on snow and getting cold feet.
So they may genuinely need more space per bird in the coop in Michigan than they did in South Carolina.
Yes, a run with a roof to keep snow out, and a layer of straw or wood chips in it, could entice them outside. Some chickens will come out, some will not--seems to depend on the individual chickens, and varies among different owners. Some people say all their chickens want out every day, some people say theirs won't go out at all. I cannot predict what yours will do, so you may have to adjust plans a bit based on how your birds behave, after you have them.