Scholar Post-doc program is an option for final year doctoral students for a 1-3 year opportunity overseas, so it can be for recent alumni, but mainly for students in their last year of their doctoral programs.
About Fulbright:The Fulbright Program is the flagship international educational exchange program sponsored by the U.S. government and is designed to build lasting connections between the people of the United States and the people of other countries. The Fulbright Program is funded through an annual appropriation made by the U.S. Congress to the U.S. Department of State's Bureau of Educational and Cultural Affairs. Participating governments and host institutions, corporations, and foundations around the world also provide direct and indirect support to the Program, which operates in over 160 countries worldwide.
If you are a U.S. citizen or permanent resident who is earning a Master's degree or Ph.D. degree, you could be one of more than 2,200 students who receive a Fulbright grant each year to go to one of more than 160 countries worldwide. It awards grants in nearly all fields and disciplines, including the humanities, social sciences, STEM (Science, Technology, Engineering, or Math) fields, and creative and performing arts.