Greene County’s Unemployment Rate In June Was Up Over Last Month’s Numbers. According To Figures From The Tennessee Department Of Labor & Workforce Development, The County Had A Rate Of 5.7 Percent In June, Which Is Up From 4.1 Percent In May. The Rate A Year Ago Was 6.7 Percent. The State Unemployment Rate For June Was 4.1 Percent And Remains Unchanged From The May Rate.