There are 3 US Post Offices in Monroe County serving a total of 10,779 residents equating to 3,593 residents per post office. It's estimated that approximately 17,204 packages are handled across the 3 county post offices annually. Monroe County is a county located in the U.S. state of Kentucky. As of the 2010 census, the population was 10,963. Its county seat is Tompkinsville. The county is named for President James Monroe. It is a prohibition or dry county."