As it is right now, there's not quite enough room to flip one of the rings over and sandwich it between two others. The middle ring forces a small gap between the other two. I suppose if you only had the three rings(but not four or more) you could get rid of the gap, but the alignment isn't quite right. The studs are just a little bit too tall.
I think I understand what you're saying about saving parts. If you let the snot walls do some of the work of holding the thing together and/or covering holes, then there are all sorts of possibilities. You say three rings instead of four, how about two rings instead of three? It's easy to get the studs lined up if you just have one layer.
My goal here was to get a structure I could cover completely with 1x1 tiles, which was inspired by this guy
, especially by his comment over here
. At this point I know what I want to do, but putting my hands on the parts is the problem.