I imagine you have a lesser rain amount than we. We have, on average 44" rain per year and thats enough to rot the big round bales , or else , at least leach out their nutrient value.
We do have a lot of the AMish use these plastic silos. These are those huge rollout plastic baggies into which they shovel "greenchop" and green hay to let it ensilate. It works pretty good and many dairy farmers use the method when they try to keep another loafing area in use all year.
We have a new system that stacks about 100 bales in a frame . We bale the hay and dont use a kicker. Instead the hay drops out on rows and my neighbor , who has one of these bale frame stackers, charges me 50 cents a bale to pull em off the field and put up in the barn in neat 100 bale bunches.
We get about 2000bales in the first cutting and then about half that in the second and third cutting. We dont always get a fourth cut , but this year we did.