A picture is worth a thousand words. See below.
The barn is carved into a hillside. That's why it has that huge retaining wall. I'm suggesting the lean-to be attached to the barn's second story roof and enclosed to create stables for animals. Lime green represents the hillside.
The animals would thus be on dirt, as they must.
I was planning on building the lean-to all along, but it wasn't going to be enclosed. It was going to be a carport for snowmobiles, motorcycles, farm equipment and chicken coops. Oh well. I could build an open lean-to on the barn's other side and still use it for vehicles, but its roof would block my store room's windows. I'm not sure I'm willing to do that.