Add hay/straw to run?

Our chickens have devoured every ounce of grass in their run. I let them out to free range on occasion so that they have access to grass/bugs, but wondering if we should be adding something to the run so it's not just a pile of dirt? When it rains it's mud-central and our ducks and silkies turn brown.

There are other things you can add like sand or wood chips. Hay and straw can quickly disappear in a wet run, but it does give them something to scratch through while it lasts.

I like putting down hay, we put it down all winter for standing on and supplemental fodder, I also use leaves in the fall and grass clippings, and shavings from the coop will get added as well, anything organic will break down over time, deep litter system.

I throw anything organic in the run, and by organic I mean plant based. Wood chips are good for managing mud. Straw is okay, it tends to mat on the top and if you try to walk on it you can slip on the mud underneath. Hint--don't fall down in the chicken coop . Not only is it gross, but if you happen to hit your head, they'll eat you alive

Grass clippings, leaves, sometimes stall cleanings all go in my coop and run. It all builds up and helps manage mud and gives the birds something to do. Toss a handful of scratch or oyster shell out there and they'll happily exercise those leg muscles for hours searching for grains. Then, every fall you can clean it out, put it on your garden, and start over. All those extra veggies you'll grow with the great compost can go back to the birds, and they'll poop more......

Here's a great description of how to manage organic 'bedding' in a run...and there's a great video of what it looks like.
https://www.backyardchickens.com/t/1037998/muddy-run-help-please#post_16017992

IMO, bare ground, be it in the garden, or in the chicken run is unhealthy soil. By adding a covering, it protects from erosion, and protects the beneficial organisms so they can work better. A deep litter in the run will encourage lots of tasty worms and insects to move in, to feed those chickens and give them plenty of stuff to do! My garden and run devour every bit of organic material my yard and flock produce. So, I go looking for extra stuff to bring in. Every trip to the dump, after dumping my trash, I load the truck up with cardboard. I have wood chips delivered by the 14 cu. yd. chipper truck. I have 2.5 loads left to spread. This weekend, I plan to go to the dump, time allowing, and park my truck at the drop off site for compostables. 1/2 hour of parking there, should yield a full load of already bagged leaves. I'll hope to get at least 3 loads to bank around the coop and green house, put into the run, onto the garden. Leaves are black gold.

In the run: when you clean the coop, toss those shavings into the run. Mow the lawn: toss the grass into the run (assuming you don't use insecticides or herbicides). Leaves: in the run. Kitchen, garden scraps: into the run. Aim for a 6" layer of organics in the run. Good luck with that! Those beneficials and the chickens will devour the stuff. What the chickens don't eat will just melt into the soil!

Thanks everyone! All great ideas!!