The Rays selected Curtiss's contract from their alternate training site Sunday, Neil Solondz of the Rays Radio Network reports.
Curtis will assume the 40-man and active roster spot of catcher Kevan Smith, who was designated for assignment in a corresponding move. The 27-year-old has yet to make his 2020 debut, but Curtiss previously made 18 big-league appearances with the Twins and Angels over the last three seasons.