I suspect your button is not Military, because I couldn't find it in the Military sections of the book "Record Of American Uniform And Historical Buttons" by Alphaeus H. Albert. I also searched through every button in that book's State Militia Regiments sections. I think it is more likely to be either a School button or what the Waterbury Button Company website calls a Career Uniform button (such as hotel employee, railroad employee, etc.).
As Duggap showed, the Jacob Reed firm was in business from 1824 until 1980... a total of 156 years. Therefore, the firm's name changed as its original founder and then his sons passed away. The same changes occurred in the Horstmann company's long timeline.
W.H. Horstmann
Horstmann & Sons
Horstmann Brothers
Horstmann Company
Because your button's backmark says "Jacob Reed's Sons" it was most probably manufactured after Jacob passed away, in the late-1800s. Also, the backmark is written in "plain block" letters, which is rare in backmarks until the late-1800s. Lastly, your backmark abbreviates Philadelphia as "Phil" instead of "Phila". The "Phil" abbreviation is seen on Reed buttons from the early 20th-Century.