There are 3 US Post Offices in Hamilton County serving a total of 2,757 residents equating to 919 residents per post office. It's estimated that approximately 4,400 packages are handled across the 3 county post offices annually. Hamilton County (county code HM) is a county located in the U.S. state of Kansas. As of the 2010 census, the county population was 2,690. Its county seat and most populous city is Syracuse. The county was founded in 1873 and named for Alexander Hamilton."