Crockett, 23, appeared in two games since getting recalled on Thursday, throwing 1 1/3 scoreless innings in that span while allowing one hit without a walk.
He began the season on the Opening Day roster and made three scoreless appearances before being optioned to the Minors on April 12.
Crockett initially struggled upon that first demotion to the Minor Leagues, allowing 10 earned runs in his first 5 1/3 innings, but he had since smoothed things out, posting a 2.25 ERA and a 17/3 strikeout-to-walk ratio in his 16 most recent Triple-A innings.
The southpaw from the University of Virginia, who was drafted in the fourth round in 2013 (No. 111 overall) made his Major League debut last season and owns a career 1.60 ERA in 33 2/3 big-league innings.
• The Indians signed four more Draft picks, agreeing to terms with RHP Matt Esparza (14th round), outfielder Todd Isaacs (19th round), infielder Sam Haggerty (24th round) and infielder Connor Marabell (25th round).
August Fagerstrom is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.