This is a pretty good article that might prove helpful.
Pat’s Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/web/viewblog.php?id=1642-fix-a-muddy-run
There are two aspects to this. First, keep as much water out as you can. It sounds like you’ve already done a lot of this, but in a good rain water seems to find it way in no matter what you do. Do you think putting a berm or swale on the upper side would help divert rainwater runoff?
The other thing is to get water out when it finds its way in. Again, you’ve set yourself up for a bit of success in that since it’s on a slope. At least it’s not in a low spot where water cannot drain. But some soils (especially clay) hold moisture and the chickens will dig holes for their dust baths that holds water, even on a slope. Egghead is right, sand is a great soil for draining but clay is awful.
There are a few problems with sand though. It disappears over time. It will sink down into the mud over time due to gravity. It’s denser than clay so the grains will sink when it’s mixed, like when the chickens scratch. You can help yourself there by putting a layer of gravel down first to for a bed and keep the sand from mixing as much. I suggest pea gravel or something round like that, maybe gravel from a riverbed or streambed. Sharp gravel might cut their feet when they’re scratching. It doesn’t happen all the time but I consider rounded gravel a reasonable precaution.
The other problem with sand disappearing is that it can wash out in a heavy rain or the chickens will scratch it out of the run. They do like to scratch. What I suggest there is to put a barrier a few inches high around the bottom of you run to hold the sand in, especially on the downslope side. If I were buying something I’d probably get landscaping cloth, but if you have something else you want to use either because it’s free or because it looks better, that’s up to you. Someone on here recently had a bunch of free bricks laying around that were priced right. Just make sure it can drain. Then build the sand up in the run so it is higher than the surrounding soil so the water has some place to go. You don’t want to dig out a bathtub in your clay that holds water and just put a bit of sand in it. That water won’t go anywhere.
Good luck with it. It sounds like you are in a situation where you can successfully manage this on a slope like that.