The button showing a droop-necked eagle is a US Marine Corps button,
Time-dating your no-backmark US Marines button:
You say it is bigger than a civil war eagle button. The largest pre-20th-Century US Marines button was 23mm. So, if your button's diameter is larger than 23mm it is from the 20th-Century.
US Marines buttons had a "black finish" on them from the World War One era to 1923. I can't quite tell whether your un-cleaned button has some "black finish" remaining in the horizontal-lined background, or not. If it doesn't, it is most probably post-1923.