The Red Sox's right fielder was out for the second straight night on Thursday, and that could extend into at least part of the weekend series against Tampa Bay.
"I really don't want to miss six weeks at a time," Drew said. "I had an injury like this [in 1999], just kind of had a little twinge in my quad like that and played the next day, and ended up blowing it up and missing six weeks. I've been in there getting some work on it, trying to keep it under wraps and just see how it goes."
Drew progressed to jogging before Thursday's game and hopes to play again in the next couple of days.
"Hopefully, at some point this weekend I'll be able to sneak back in there to kind of gauge what I do during the game," Drew said. "I've got to play it smart. If I hit a ball in the infield, instead of trying to get an infield single out of it, you've got to realize it's just going to be an out."
With Drew out of the lineup and Jacoby Ellsbury (sore right groin) also out during this series against the Blue Jays, Sox manager Terry Francona has been able to get more of a look at Brandon Moss, who already made his mark on Opening Night in Tokyo with a game-tying homer in the ninth.
Moss was recalled from Pawtucket on Saturday when Sean Casey was placed on the disabled list.
The Red Sox like what they see in Moss, who is expanding his horizons by also learning how to play first base.
"He's really interesting," Francona said. "Even Mikey Lowell was saying when he was there that Mossy stands out a little bit. The ball comes off his bat like a Major League hitter -- even when you watch him in BP. He's an interesting player. He's a good outfielder. [I] actually can put him in center field, which I don't know if that's his everyday position, but you can put him there. You can put him at the corners.
"Can he play first base? It's intriguing. Because all of a sudden, if you have a young left-handed hitter who can play three, four positions, that's pretty valuable."
Moss nearly had a game-winning hit on Wednesday, but pinch-runner Jed Lowrie was thrown out at the plate. It wound up not mattering, thanks to Jason Varitek, who followed Moss with a walk-off single.
Ian Browne is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.