I don't think that storing the hay that has been previously stored outside for a winter will keep all that long. Yes, it is better than nothing. But if it has already gotten wet, and has some rot on the outside, it will continue to deteriorate. If you are wanting to store hay for any length of time, it needs to be stored on the flat side so bales retain their shape, on a dry gravel type bed or pallets, stored under cover BEFORE it has a chance to get wet. So, within a few days after baling it needs to be inside, under cover. It will keep for several years with little breakdown. It will get a bit drier as time goes on and lose some nutrients, but it won't rot as there is no moisture to start the process. We have had small square bales in the barn that have been there for 5 years, and they are rather dried out, but still smell pretty decent, some rodent damage to strings etc., and we try to rotate out the oldest but the cows will eat it pretty good if it was decent going in the barn.
If it has already been wet, with some rot around the outside, it will continue to break down. You might get 2 years out of it stored, if now in the dry, but it will be slowly "rotting" due to the process having already started. Wrapping may actually cause it to rot faster with the summer temps/sun causing condensation. Anaerobic bacteria will be working in the wrapped hay,
aerobic bacteria will be working in the unwrapped hay stored under roof cover. We have gotten and fed 3 year old hay stored outside and it has some rot but the cows have often eaten it pretty well. If we are given hay like that because a farmer wants it cleaned up out of the windrow/fenceline, we take it as we are putting nutrients back into the soil as well as organic matter. If net wrap is used, they will stay together for transport better, but they will still continue to break down.
We try to feed our "old hay" first, feeding say 2 old to one "new" , so they are getting some better nutrition from the "new hay" but are getting more of the bulk and fiber from the old also.
It will also depend on the type of hay. We got some old rye hay that had been baled off a field that a farmer had been renovating that was 3 years old. Looked like rotted crap. The cows devoured it.... Go figure. They had good current hay and we were spreading this stuff around out in the field and they were practically licking the ground to get every last little bit.