Most people who store their own hay in at least moderate quantities tend to have some rotting hay in their shed/barn. It tends to get pushed to the back and sits there for years--they can't feed it to their animals and it's a hassle to get rid of it. Luckily, rotting hay is a great mulching medium. For most it would simply be a matter of asking if you can haul their bad hay out for them, and if they have a larger amount they might even consider paying you for it.
The drawback, and another reason people are so happy to be rid of it, is this hay often smells bad. You should wear something to filter the air you breathe and take a pitchfork--it might be loose hay. If you hit the jackpot and find round bales of rotting hay you'll need some way to load and transport it.
You need to pick the hay up when the barn is at it emptiest. It will be next to impossible to navigate around a filled barn to get to the bad hay. Hay tends to be used in the cold months and the barns are refilled at harvest, which is around May/June here in central Texas. But you can scout around anytime and ask people when the best time to come would be--they might want you to get it now!
Some people discourage the use of hay in mulching, saying it is filled with seeds. If hay has been grown and harvested properly there will be no seeds. The harvester wants to cut hay before it has gone to seed. When hay seeds it sends nutrients up into the seeds that have been building in the stalk, thus if it's already seeded most of the nutrients are in the seed head and the straw is mostly nutritionally useless. Many people selling hay don't care about this and will instead try to get the tallest hay so they can make more bales to make more money. However many people who grow and harvest their own hay will be more mindful, so it might be worth it to find people who grow their own.
Having said that you may end up with some seedy hay. You can look at the hay and tell if it's saturated with seedheads or if there are just a few here and there. I've sheet mulched this year with hay that had a few seeds. Green shoots come up now and again through my top layer of compost. About twice a week I'll pull them out by their roots. For around 1000 sq ft this takes less than five minutes.