We store all our round bales outside. We shoot for having approx 100-200 left over every year so that we have a cushion.Yes, they lose some quality. Yes there is some "waste" as the outside deteriorates. All the "old" hay gets fed first to dry cows. They need the lessor amount of nutrition and it is better to not overfeed a high quality hay that will sometimes cause the calf to put on exceptional growth and size before being born. But it has been studied and proven that the dry cow does not need the same quality of feed as a recently fresh lactating cow.
Depending on what the hay is, we have cows that practically lick the ground when we have fed some old hay. Stuff that you think is "junk" they are fighting to eat. Go figure.
On the older hay, with the outside couple of inches weathered, we normally roll it out so that they have something to lay on besides the cold and/or wet ground. It is putting organic matter back into the soil so it is not really wasted. Sure the cows may not be doing more than picking through it, but you are still getting those nutrients added back into your soil. If you feed it all in feeders, or in one place and don't roll it out, then take and load the accumulation into a manure spreader and spread it on the fields, you are still putting nutrients back into the ground.
We have some 3 yr old hay that was part of the left over when we rented the big farm. We feed that out with "good hay" and let the cows pick through it. They will find what they want to eat and the rest goes back into the soil. Anything that rots back into the ground is feeding the bacteria, and micro organisms that make up the basis of the soil nutrients. If the hay came from fertilized fields, then you are is essence putting that fertilizer back into the field or pasture where you are feeding.