Right-hander Jake Petricka was recalled from Triple-A Charlotte and will join the team in Kansas City.
Jones was slowed at the start of spring training with a left gluteus muscle strain. He missed the first week of Cactus League games but managed to make eight appearances in Arizona.
His bid for the vacant closer's job fell short as Matt Lindstrom won the job despite dealing with his own oblique strain. Lindstrom blew a save opportunity Thursday and took the loss in the series-ending defeat to the Minnesota Twins.
Jones appeared in two games in the season-opening series, facing five batters without recording an out. Four of the five baserunners he allowed scored.
Petricka did not appear in Charlotte's season-opening victory Thursday. He pitched in 16 games with the White Sox last season, gaining major league experience for the first time. He had a 3.26 ERA in the big leagues and struck out 10 batters. His major league debut came Aug. 22 at Kansas City.