The M.C. Lilley & Co. maker's-mark means the hat dates no earlier than 1870, when that company was founded, so it cannot be a civil war era hat. That company lasted into the 1940s. It dealt mostly in "fraternal society goods."
Although the chinstrap button is a US Army one, the embroidered emblem is a Fraternal Society symbol. (I've seen it before, but at the moment can't recall which Society, might be Knights Of Pythias.) Also, the white cord across the cap's top is not military.