Supporting Americans Wrongfully or Unlawfully Detained Abroad Act of 2023 This bill authorizes travel assistance and mental health support to U.S. nationals wrongfully or unlawfully detained abroad and to certain family members of such individuals. For wrongfully or unlawfully detained U.S. nationals, the bill authorizes the Department of State to provide compensation and assistance if needed for return travel to the United States after their release. The bill also authorizes mental health support for such individuals and certain family members of such individuals. For certain family members or representatives of such detained individuals, the State Department must provide financial assistance to cover travel and lodging expenses for meeting with the U.S. government if (1) such financial assistance is requested, and (2) the family members or representatives live more than 50 miles from Washington, DC.