Red Sox manager Bobby Valentine expects Hill, who was placed on the disabled list on Sunday, to miss about "a month or more."
"Rich Hill has a strained flexor muscle, and he'll have some time to rest that and get it back," Valentine said. "He threw 94 mph his last game and threw his breaking ball and felt a little stiffness. That stiffness was not in the ligament, luckily. They feel it was in the flexor muscle."
Before Hill's injury, the Red Sox had the luxury of going with three lefties in the bullpen.
Now they have Franklin Morales and Andrew Miller. Morales, in particular, may be used in some different spots to help fill the void left by Hill.
"I'm trying to figure that one out," said Valentine. "I think he knows that I explained a little plan to him. One of the things that was mentioned is that he's very durable and flexible. I want his mind to remain open."