Chickens as a rule, much prefer shade to sunshine, unless it's cold, then they might sun themselves a bit.
As far as ease of building, and cost, it's real hard to beat a hoop coop for your run.
If you go to the article section of this website, there's lots of examples of hoop coops. You're taking a 50 inch by 16 foot cattle fence panel (about $22 each at
Tractor Supply), and setting each end on the ground about 8 feet apart.
That produces a hoop arch about 6 feet tall, and as mentioned, 8 feet wide. Put four of them together, and you've got an 8 foot by 16.5 foot run, excellent for 12 or 13 birds (10 square feet each), that with half inch hardware cloth, wired or zip tied over the cattle fence panels, along with a wire anti-dig skirt around the perimeter of the hoop, is absolutely predator proof.
Less panels for less birds, more panels for more birds. Then you can add a 16 foot wide tarp over it. In the summer, roll up 2 feet on the sides, that will leave 2 feet of open wire along the sides of the hoop for complete ventilation, in the winter, seal up the sides.
Most people will use a 2x6 frame on the ground, but some just drive a couple of metal fence posts on either side of each cattle panel, and wire the cattle panel to the metal post, so it can't move.
I'd attach it to your coop, on your coops south side. The arch of the hoop, should face south, to southeast. That'll take advantage of the low winter sunlight, on the hoops flat open side. You'll be able to leave this side (the south side, open wire all winter, don't cover it. Try to fit your door at the north end of the hoop. It might not be a bad idea to offset the hoop from the coop, giving you south sun on the coop, and giving you room to get the door into the hoop on the north end. But if that's not possible, and your coop takes up the whole north end of the hoop, you could put a door, with open wire, on the south end of the hoop.
My hoop coop is a chicken tractor, here's the open south end. It's the same principle as a stationary hoop coop, except yours doesn't need to be as heavily built as this one, which gets dragged 8 feet sideways every week.
This is the North end of my hoop. It's the only side that's plywood, it has a homemade open wire door, and a rollout, through the wall nesting box.
The reason I'm trying to keep everything out of the south face of your hoop is to preserve as much openness as possible for winter sun, and so you could put a perch across the whole south end of the hoop. My birds love sitting at the open end of the their hoop, enjoying the breeze, and watching the world through the open wire.
Make sure the north side (against your coop), the east side, and the west side, are all sealed using the tarp, and coop, during the winter. That'll create a still cushion of air in the run, even with the south end completely open.
The open south end of the hoop provides plenty of ventilation during the winter, with even more in the summer when you roll up the side tarps a couple of feet.
Like I said, you can also roll the sides up on the tarp a couple of feet during the winter, if you get a few good days of nice weather, and leave it rolled up the other three seasons.
The skirt is very easy too, just one inch by one inch welded wire laid out two or three feet along the outside perimeter of the run, staked down with landscape staples (cheap and easy to find). In no time the grass grows through the wire, and you won't even know its there, mow over it with the lawn mower whatever, buti it'll prevent predators from digging under your fence, and getting your chickens.
That's a very inexpensive, absolutely predator proof run you can stand up in, that allows you to easily shade and waterproof the whole thing. Cover the whole thing, plenty of sunshine will come in the south end in the winter, and the east and west sides the other three seasons, when you roll up the tarp.
Cold never bothers chickens as long as you keep the rain and wind off them. This system does that beautifully, and as inexpensively as is possible, for a system that's predator proof.
Sorry.
