I guess I'm kind of old school. I don't know what the deep litter method is, but I may go look it up. If it was me, they would get hay or straw or chips in the "house" part of the coop. I'd use whatever I had on hand or could get cheaply, I personally don't think that the chickens really care since they roost over the stuff at night. Same thing goes in the nesting area, they will use what you give them. Some folks will say don't use hay or straw but I have not had an issue with it and I've been doing for a LONG time. In the run area I would not lay anything down. Let the ladies enjoy the grass. Once the grass is gone move it over if possible. If not then I would only lay something on the ground if it gets poopy. Keep in mind you may get eggs laying on the ground if you start to do that. And not to state the obvious but clean out as needed. Hopefully this helps.