Naughton was optioned to Triple-A Memphis on Friday.
Naughton was claimed off waivers from the Angels on Monday but won't get the chance to earn an Opening Day roster spot for his new team. He could return to the big-league roster to cover for an injury later in the season, but his 6.35 ERA and 12:14 K:BB in 22.2 innings last season suggests he still has some development to do if he's to become a weapon at the highest level.