My chickens are free range from sun-up to sun-down on 50 acres. The only bare dirt over the whole farm is in the barn stalls and the chickens seldom choose to spend any time in there. However, as someone else said, I clean the coop and dump the bedding out on the ground for the chickens to scratch about in and they love it. They will ignore the entire farm in favor of picking through that pile of bedding. They go through it pretty fast though, so this is only a short-term solution (at least until you've accumulated enough to create a deep layer). Interestingly, they don't pay the bedding any mind while it is in the coop.
I would suggest layering as much material as you can into the run. Grass clippings, raked leaves and even bagged wood shavings are perfect for this. If you give them vegetable scraps from the kitchen, bury what they don't eat loosely with the layered materials to attract insects and worms the chickens can hunt for. As kelbycbf suggested, a few logs will also attract insects. I do this in my grow out pen with the oldest, most rotten logs I can find on the farm because the chickens will also peck at the log itself to find the insects that are inside it helping it to decay.
You can also hang bunches of vegetables like celery or lettuce in the run so that the chickens can 'jump' for it to get a bite. This entertains some of mine quite well and doesn't interest others at all. I sometimes cut a watermelon in half and let them eat all of it that they want, then I turn the empty half upside down on the ground - insects accumulate under it super fast.
FWIW, I have no problem using hay. As long as you keep it loose and not wet and mucky, it's perfectly fine. My chickens love to explore the 300+ hay bales stacked in my barn. They've created tunnels all through it. I can only assume that they do this because they can find treats in them. I've never seen a mouse or rat in my barn, perhaps this is why.