There are 5 US Post Offices in Greene County serving a total of 8,398 residents equating to 1,680 residents per post office. It's estimated that approximately 13,404 packages are handled across the 5 county post offices annually. Greene County is a county in the U.S. state of Alabama. As of the 2010 census, the population was 9,045. making it the least populous county in Alabama. Its county seat is Eutaw. Its name is in honor of Revolutionary War General Nathanael Greene of Rhode Island."