Welcome Tigeroak
I think your plan to add the lean-to on to the green building is a good one, hope you get it done sooner than later. What I see you have happening now is a common mistake many people make when stacking wood. You have a lot of wood stacked right against the wall of that green metal building directly under the drip line off the roof. That creates a compound problem, first is that all the rain/snow that lands on that roof runs off the edge and lands directly on the wood stacked against the wall. Second problem is that once that wood leaning against the wall gets soaking wet it would have a lot of trouble drying because the air can not circulate around it because it is so tight to the wall.
I have seen a lot of cases where people do this (stack wood under the drip line of a roof), the one thing you have going for you in this case is that the wall is made of metal and won't rot. Many times people will do the same thing up against a wood structure wall and then not only does the firewood stay wet and begin to rot, so does the wood sheeting or siding on the wall.
Having a lean-to roof off that wall would be a great place to store your wood.