The fibrous paper backing Fyrffytr1 is talking about is a circular disc inside some 3-piece buttons. I've seen it inside some excavated (dug") New York State Militia 3-piece buttons and a US Staff Officer button, which were corroded badly enough that the applied rim came off and the front and back came apart, revealing the fibrous paper disc inside. No idea what its purpose is.
BigCypressHunter, the 2-piece early-1900s button you're remembering is one version of "false 3-piece" (a.k.a. false staff) button. The rim of the front is wide and flat, made to resemble the applied rim of a 3-piece "Staff Officer" button.
Having very closely examined the photo of the front of ripvanb's button, I have to say it looks like the front of a 3-piece button whose applied rim has corroded/broken off. (I've seen many 3-piece buttons in that condition during my 40+ years as a civil war relics digger & dealer.) I'm pretty sure it is not the 2-piece button shown next to it in vpnavy's post. But all of that being said, I could be wrong. It would be helpful to see a well-focused closeup "sideview" photo of ripvanb's button.