Matt, these U.S. Army "Great Seal" general service buttons were introduced in 1902 and have been in use ever since. Some can be dated by the backmark, but often that's not much help either.
I found one in excellent condition about three weeks ago near Ft Bayard where the buffalo soldiers were stationed for several years pre 1880. A large button, and design looks the same. On the back is "steele&johnson". The ones without "mfg co" are supposed to be pre 1875. That would be about the right time frame for the most active Ft Bayard military period. It seems that the Great Seal has been used since at least the war between the states.
Steele & Johnson were still making buttons in the early 1900's. I believe the firm ended around 1920, when it was bought out by a competitor, the Waterbury Button Co. Although the Great Seal as an official U.S. symbol was introduced in 1776 and finalized in 1782, its use in the design shown on U.S. Army General Service buttons was not authorized and did not occur prior to 1902, when the following directive was issued: "One style for all arms and all grades, private to Lt. General. The only exception is the Engineer Corps, who are to retain their present design. Bronze for service and gilt for dress. The design is a copy of the great seal of the U.S." (Source: Record of American Uniform and Historical Buttons by Alphaeus H. Albert.) Even though they may have been found at sites where there was earlier military occupation or action, there is absolutely no possibility that these buttons predate 1902.