"Overnight, Ian made some progress, but hamstrings are tricky," Black said. "This is the best thing to do for Ian. Now we can make sure the leg is 100 percent."
Kennedy left the game after 33 pitches, having allowed two hits and two walks with two strikeouts in 2 1/3 innings. He sustained the injury in the second inning on a pitch to Joaquin Arias.
"I just felt something grab and I could feel it slowly and slowly getting tighter," Kennedy said Thursday.
Odrisamer Despaigne, who replaced Kennedy and retired all 14 hitters he faced, will slide into Kennedy's spot in the rotation and will pitch Tuesday against the D-backs.
Despaigne was 4-7 with a 3.36 ERA in 16 starts for the Padres in 2014.
"We're going to miss him, but injuries are part of the game," Black said. "This looks to be a short-termer as far as injuries go."
The team purchased the contract of left-handed reliever Chris Rearick from Triple-A El Paso and selected right-hander Brandon Maurer to give them some bullpen depth.
Reliever Nick Vincent was optioned to El Paso, though he'll be back. He had pitched in three of the first four games and the team needed fresh arms.
Corey Brock is a reporter for MLB.com. Keep track of @FollowThePadres on Twitter. This story was not subject to the approval of Major League Baseball or its clubs.