Because you requested a more-specific timeframe than "about 1870 through the first 1/3rd of the 20th-Century," I'll do waht I can to narrow down that timeframe for you. Your 3-piece GAR button has an iron back ...which is quite rarely seen on 3-piece buttons. A man named Goodwin received a US Patent for iron-backed 3-piece buttons on July 27, 1875. Some iron-backed GAR buttons have a backmark with his name and that date (or an 1878 date).
Also, as decades passed, the number of surviving civil war veterans dwindled, of course. Therefore, far fewer GAR buttons were manufactured after 1900 than in prior decades.
So, putting all of those pieces of information together, the statistical odds favor your ironback 3-piece GAR button being manufactured sometime between the latter 1870s and the latter 1890s -- although of course it could have been worn and lost later than that time-period.