(My grandma was blown up, a story she loved to retell when I was a child... "Did you know I was blown up? Threw me 50 feet from the barn!")
Would Romanian haystacks help? Or is it impossible to do it on a large scale without a large amount of manual labor? Can you just drape a tarp over each hay bale?
That said, I’ve seen farmers use old billboards as tarps in the past. Otherwise, it’s usually either a barn or a plastic sheeting.
All hay is irrigation intensive, but my understanding has long been that premium quality alfalfa hay is especially so. The fields around here used to be flood-irrigated, but as the cost of water has risen over the past 10-15 years, I have seen that practice curtailed drastically. You lose too much water to evaporation. That implies heavy investment in more precise irrigation systems.