There are 1 US Post Offices in Hancock County serving a total of 5,031 residents equating to 5,031 residents per post office. It's estimated that approximately 8,030 packages are handled across the 1 county post offices annually. Hancock County is a county located in the U.S. state of Tennessee. As of the 2010 census, the population was 6,819, making it the fourth-least populous county in Tennessee. Its county seat is Sneedville."