There are 21 US Post Offices in Hamilton County serving a total of 358,186 residents equating to 17,056 residents per post office. It's estimated that approximately 571,705 packages are handled across the 21 county post offices annually. Hamilton County is a county located in the U.S. state of Tennessee. As of the 2010 census, the population was 336,463, making it the fourth-most populous county in Tennessee. Its county seat is Chattanooga. The county was named for Alexander Hamilton, the first Secretary of the Treasury. Hamilton County is part of the Chattanooga, TN-GA Metropolitan Statistical Area."