How about planting climbing vines like jasmine or clematis or another sort on either side of the run. It would take awhile but they definitely would shade your run and keep a lot of the rain off the chickens.
I realize I am a little late to the party, but we just finished building our "Carolina Coop" knock-off. Someone had some "free" (if you subscribed to a monthly journal he was involved with, cost $5 a month and I forgot to unsubscribe so I paid $10!) plans that I printed out to get my husband...
