Hay doesn’t soak up water nearly as well as straw, so if it’s muddy and they’re outside, it will need extra care to be sure their bed is dry.
I got caught out even with straw one year, assuming it would be dry inside the ark, but they were taking so much water in on their legs and bellies, even the very generous straw bedding was in fact getting wet and cold. After that I used to toss out older straw from the front of the ark every couple of days to make a big mat in front of the ark, so they’d be dried off a bit before they got into the ark. I don’t think that setup would work at all with hay when the weather is wet and the ground muddy; the outside hay wouldn’t have much capability to dry them as they go in.