Seattle manager Eric Wedge said Smoak's injury isn't serious, but "it's just not a smart move to push it right now."
Wedge said Smoak will be reevaluated on Thursday.
Second baseman Dustin Ackley got the call at first in Wednesday night's game against the Indians, his first start at that position in the Major Leagues. Ackley did play first base almost exclusively for three years at North Carolina, however.
"If there was any position I'd feel comfortable going back to, that would be the one," said Ackley, who converted to second base shortly after being drafted by the Mariners in 2009. "I made a couple starts in the outfield, but pretty much when I got to college I had so many arm problems I started at first and played there probably 95 percent of the time.
"It's almost my first home, but I haven't been there for a while."
Ackley said he'd have to borrow teammate Alex Liddi's first-base glove, just as he did when he played most of a Cactus League game at first toward the end of Spring Training.
Smoak said he started feeling his hamstring tighten up on Saturday, and it continued to bother him when he was running in recent days, including his four-single night in Tuesday's 9-8 loss to Cleveland.
"It's one of those things where I talked to Wedge about it and I'd rather miss a day than a couple weeks," Smoak said. "That's where we're at right now. It's day to day. Nothing serious. The goal is to get treatment, calm it down and be ready to go tomorrow."
Smoak said the leg doesn't bother him when he's swinging the bat, but only when he's running the bases, which happened four times Tuesday as he raised his average to .250.
"You get some hits and you want to stay in there," Smoak said, shaking his head. "You start feeling better at the plate, you don't want to come out. I tried my hardest to play today. It's just one of those things where it's early in the year and I don't want to miss a lot of time."