Rays manager Joe Maddon said Floyd hurt his right shoulder in Game 2 at Tropicana Field either while trying to slide or on a pickoff play at second base.
"Yesterday he was hurting, to the point where it was difficult to sleep," Maddon said. "We tried to get through last night with him. He was available, but he's not today. We've got the appropriate paperwork in order, Hinske's on and Floyd is off."
Floyd battled injuries during the 2008 season but still managed to hit .268 with 11 home runs and 39 RBIs in 80 games. In the postseason, the veteran slugger hit .222 with a home run and two RBIs in 18 at-bats.
Maddon said it's unlikely Hinske will make any starts at designated hitter, but said he could be used at first base, third base, right field, left field and as a pinch-hitter, which could make him a valuable player if the need for a double-switch occurs during one of the remaining two games in Philadelphia where National League rules are used.
"It's kind of bittersweet, you never want to see anyone get hurt," Hinske said. "But I'm excited to be active and have a chance to help any way I can. So I don't know if it will be pinch-hitting -- I definitely have a chance to pinch-hit in a National League game -- so whatever I can do to help, it will be fun."
Hinske was on the Rays roster for the American League Division Series but did not play. The Rays then elected to go with 11 pitchers and 14 position players for the American League Championship Series and the World Series, and Hinske was the odd man out.
"It's always tough as a competitor [not to be able to play]," Hinske said. "But the bottom line is winning, and we have been winning. ... I've been around the game long enough to know it's a business. It's not about one player. It's unfortunate what happened to Cliff."
Hinske, 31, made the big league team out of Spring Training as a non-roster invitee and went on to hit .247 with 20 home runs and 60 RBIs in 133 games.
Despite his recent inactivity, Hinske said he felt good.
"I've been taking batting practice," Hinske said. "I haven't had an at-bat in about three weeks, but if I get in there, I'll swing. I'm going to take a whack for sure."
Hinske has been the consummate professional throughout the season, the postseason in particular, and in the way he handled the news about being left off the roster. He said he began to get an idea something might happen on Saturday.
"Cliff kind of told me that he's been hurting," Hinske said. "He just told me to be heads up. But that was it. Joe kind of gave me an [idea] about it yesterday, 'He might not be able to go, so be ready.' When I got to the field today, they told me."
Bill Chastain is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.