The straw that I buy is short and flat, almost like wide bladed grass blades yet it is labeled straw for whatever reason. It's more leaves than stems.
I do deep litter in my coops but I usually have more grass clippings, leaves, pine needles, etc than what I need in the two coops so that goes into the run as well. Since I started doing that beginning this past summer I haven't had issues with a mucky coop.
I'm going to look around and see if I can find a bale of grass hay like you are talking about. I think they would like it if I just put it in the run and let them tear through it and spread it, since I don't have anything to add to the coop this time of year.
It works very well. Its what I do too, and include little things like small sticks and pieces of pine cones. Basically whatever I can collect with the sweeper on my riding lawn mower.
I think it's called mulch hay or construction hay, might make it easier to find

And wow that seems like it works well!
Do you think it works with a floor though? Mine has i think 3/4 inch pressure treated plywood with scrap linoleum/vinyl flooring on top. I was planning on just having pine shavings on the floor and changing it out every so often (with the new coop it probably wouldn't need to be changed as much) but idk if that would get expensive or something?