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They look like thick dust on the barn floor.
Are you sure these are hay mites? We always get straw mites from the bedding on our cattle and donkeys in winter. But these are easily treated with ivomec (or similar) pour on. The animals scratch from them before treatment, but are unaffected after. But there are never so many mites that you can actually see them, and they've certainly never crawled onto me.
However we do get other mites that live in our disused sheds in the dust. These look like a silvery grey dust and will crawl onto you if you inadvertently touch a surface that they are on. I spray every now and then with permethrin (ie crovect or spot on) diluted about 10x with red diesel, or spread some louse powder about. This gets rid of them for a while.
I would suggest you spray or put louse powder on your barn floor and then sweep or vac it all up before putting some fresh hay in. See if that sorts your problem.