OK, I'll start my "show up" right here with Julie friend!!
I've found that here in NC, it doesn't seem to matter - this year everything is "sweating" or constantly "dewy" with the humidity -even when we don't have out & out rain (which we've had an AMAZING amount of this year & not just from Hurricane Matthew!).
They do carry heavier tarps at both Lowes & and TSC. They are commensurately more $, tho.
Here is the king of tarps website. They, too, are higher - not sure if it wouldn't be less to find tin for a roof - which you can also attach to the hooped coop panels.
http://tarps.com/My "better" hay suppliers actually utilized more open barns. They have roofs on them, but the sides were either completely bare or had wood slats up with spaces (not a solid wall but like a pallet). Then, when folks came to get horse quality hay - they never pulled from the outside row next to the wall (weathered) or on the bottom sitting on the pallet. I've always appreciated that -even though when we get the bales they go into the field and get "stored" where they land, LOL. I found that all the hay, even when on pallets, seems to get damp-ish or moldy if the humidity high or lots of rain (or like this year - BOTH) on the bottom pallet. In our situation, we've been able to pull apart and throw moldy or ?able hay out in the various pastures and let them pick thru it.
In the late 90s early 2000s, before we moved up to Lillington and now to Cameron, all of our hay suppliers seemed to know which type of seed/hay they had. A couple even gave me choices and often I'd have 2 or 3 kinds of hay rounds sitting in our pasture(s) - all coastal bermuda - and the ponies would eat what they wanted. When I moved to Lillington - all of a sudden none of the hay suppliers seemed to know that they got a certain "type" of coastal bermuda. And when asked ".... well, it's coastal..." scratchin' their heads at my ?s. Some of the varieties, when I was further south, seemed to have better lasting qualities, though now I'd be hard put to remember which was which, not that it matters when the folk up here don't even "know" what variety seed they've planted and hayed...
Tarping hay here in NC has never worked well for me. But then, I compare it to the western states where we just didn't have the rain or the humidity, and no, u can't do that.
For your panel wrapped hay, you could put it up on a pallet first. U could even put a hooped cover over it - keeping the hay itself protected and offering "cover" to the beastie that's eating it, LOL.