When I had my coop built, I had them put two hen doors. One door goes to existing run, the other will open to future run and the plan is to alternate biweekly usage. Not sure if this will work, but I find there is a lot of mold growth in the run now from feed and poo and just general dampness (we having tons of misty cloudy rainy ick this yr). I wish we could free range them but the hawks are a plague. I will probably end up setting the culls loose when they are big enough to defend themselves from the gamefowl. For now its getting crowded.
I would suggest a bunch of sand in the run....
Also, look at those grow frames that some people use.... The grass grows up through the frame for the chickens to eat, but enough of the grass and the roots are protected, so it doesn't die.