The History of the Americans
Juan Ponce de Leon and the (Official) Discovery of Florida
We now hasten to the first European exploration of the lands now constituting the United States, and that means the first sanctioned expedition to Florida in 1513 by Juan Ponce de Leon, and the legend of the Fountain of Youth. Florida would turn out to be enormously challenging, so it will still be more than fifty years before the first successful permanent settlement at St. Augustine.
As discussed in the episode, there is some debate over Ponce's route, so the various maps available online differ in important respects. Here's one from 1913, which is as professional and on target as any that I found.
CWCID
Selected references for this episode
Samuel Turner, "Juan Ponce de Leon and the Discovery of Florida Reconsidered"
John McGrath, "Sixteenth-Century Florida in the European Imagination"
T. Frederick Davis, "Ponce de Leon's First Voyage and Discovery of Florida"