Kingham, 27, was designated for assignment by the Pirates on Saturday.
Reliever Ryan Tepera was transferred to the 60-day injured list to make room on the 40-man roster.
Kingham carried a perfect game into the seventh inning of his major-league debut last year, retiring the first 20 batters he faced before surrendering his first hit - a groundball single - to St. Louis Cardinals shortstop Paul DeJong.
Since then, the right-hander has struggled. In 34 2/3 innings this year, Kingham owns a 9.87 ERA and 5.40 FIP while making 14 appearances including four starts.
The Blue Jays employ one of the worst starting staffs in baseball, ranking 12th in the American League by FanGraphs' WAR while collectively posting a 5.02 ERA and 4.85 FIP. Twelve different pitchers have already made at least one start for Toronto this year.