"Picuda" underwent overhaul in the Charleston Naval Shipyard from 13 October 1958 to 12 March 1959, followed by a brief period of refresher training in the New London, Connecticut–Newport, Rhode Island, area. She resumed her former duties at Key West on 27 March 1959 and put to sea from that port 1 June for her third Mediterranean tour. She touched at Norfolk 4 June and reached Gibraltar on 15 June. Operations in the Mediterranean during this tour included visits to Naples and Genoa, Italy; Marseilles, France; and Lisbon, Portugal. She departed the latter port 26 August to resume training duties at Key West, Florida.