Vietnam-Era Veterans Hepatitis C Testing Enhancement Act of 2019 This bill requires the Department of Veterans Affairs (VA) to carry out a one-year pilot program to make hepatitis C testing available to certain veterans at events organized by veterans service organizations. To be eligible, a veteran must have served in the active military, naval, or air service between February 28, 1961, and May 7, 1975, and must have been discharged or released under conditions other than dishonorable. The program shall be carried out in two rural and three urban Veterans Integrated Service Networks (regional VA health care administrative areas). The VA shall expand the program to all networks if at least 350,000 veterans are tested for hepatitis C by the program's termination date.