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4/22/2014 8:02 P.M. ET

Gillaspie sits out with sore left hand

DETROIT -- Third baseman Conor Gillaspie was out of Tuesday's lineup against Justin Verlander and the Tigers and most likely will miss Wednesday's contest because of a sore left hand.

The injury was caused by the cold early weather and Gillaspie, who ranks tied for third in the American League with seven doubles, getting jammed at the plate.

"If it was an important playoff game, I'd probably suck it up and play, but I really don't want this nagging on all year," Gillaspie said. "Any time you go out there thinking more about how something hurts than your job, it's kind of like, 'I'm not really sure how much I can help tonight.'

"I'm just trying to give it another day or two and then I'll probably start wearing a pad on my hand, even though it's something uncomfortable. It's kind of something I might have to do until it gets better."

Gillaspie has been wearing batting gloves to try to combat the cold and try to give extra padding for the fat part in the middle of his hand that was tender on Tuesday. He doesn't look at the injury as anything serious.

"I've been pretty fortunate throughout my career to be pretty healthy for the most part," Gillaspie said. "If it means taking a day or two now as opposed to scuffling for 10 days straight and then going on the DL or something, I don't want that to happen. I'm just going to take it easy for a day or two and should be back pretty soon."

Relief corps showing signs of working into shape

DETROIT -- The White Sox bullpen is slowly but surely coming together.

The relief crew threw 6 2/3 scoreless innings over the last two games entering Tuesday night, and the White Sox are starting to know what they have. It was a process they weren't completely afforded in Arizona during February and March.

Ronald Belisario had his Spring Training arrival delayed because of visa issues coming from Venezuela. Daniel Webb missed time because of the passing of his mother. Nate Jones had injury issues that have carried over to the regular season. Jake Petricka wasn't even part of the Opening Day roster.

"We are getting to see them more," White Sox pitching coach Don Cooper said. "Nate is still a missing man. So, we've had to do some stuff and try some stuff and look at some stuff.

"What we were looking at early, the stuff that was happening, it was more self-inflicted: the walks and the falling behind. If we can lick that part, I think we'll be well on our way to getting where we want to go."

Cooper understands that the bullpen has been the target of early criticism, with its 5.48 ERA and 43 walks issued. He also knows that this particular group will need defensive support to help it succeed.

"We've got a bunch of ground-ball, contact guys. Certainly, we are reliant on our defense in that situation," Cooper said. "There's not a lot of strikeout guys. They can get strikeouts, but they aren't going to be a strikeout-an-inning guy, to a large degree most of them.

"I'm seeing more ground balls with [Belisario], with Petricka. We are not quite there, but we are hopefully heading in that direction."

Eaton wearing brace, return uncertain

DETROIT -- In testing his sore left knee on Tuesday, White Sox leadoff man and center fielder Adam Eaton was given a slightly different diagnosis than the original mild strain in the lower part of his left hamstring that caused him to leave Saturday's game in Texas.

"Apparently there's a tendon that is kind of rolling over on the inside back part of my knee. It's been kind of an easier fix," Eaton said. "We're making sure the tendon doesn't move and that equals no pain."

Eaton ran stairs, did some lunges and even ran some incline, during which everything went well. He declared himself at 95 percent, but still missed a third straight game because of the injury.

A return on Wednesday or Thursday in Detroit, with Eaton wearing a brace on the knee to help the tendon issues, is possible but still not certain.

"It's not a bad thing to even give my leg some more rest," Eaton said. "We'll see about tomorrow. Keep pushing it. It's something that will fix itself and heal over time. Just wear this [brace] in the meantime and continue to play later."

"He's always going to say he's feeling great," said White Sox manager Robin Ventura. "He said he was feeling fine the other day, and he wasn't. We'll see how he moves around. We'll know more by watching him and seeing how he moves, talking to [White Sox head athletic trainer] Herm [Schneider] about it, than just taking his word for it and letting him go out there."

Ventura not basing decisions solely on pitch count

DETROIT -- No direct correlation has been established between Chris Sale's career-high, 127-pitch effort Thursday night against Boston and his trip to the disabled list Tuesday with a flexor muscle strain in his left arm. There's also nothing that would stop White Sox manager Robin Ventura from using Sale or his other starters in a slightly elevated pitch count again, depending on the situation and how the given pitcher is feeling.

"You can let him go in certain types of games, I think that was one to let him go," said Ventura of Sale against Boston. "There are a certain amount of guys who are No. 1's like him and at the top of the heap as far as certain pitchers, and if you take him out you're going to hear that, too.

"So I don't necessarily worry about that part of it. It's more of how he's feeling and when he's out there and watching him pitch. That's the bigger decision to make than just how many pitches he's thrown."

Sale doesn't pay much attention to the pitch count or the time of the season, with the 127 pitches coming in his fourth start.

"I just think it's something that happened by chance," Sale said.

Third to first

• Gordon Beckham remains on track to return from his injury rehab assignment with Double-A Birmingham this weekend. Beckham started the season on the disabled list with a strained left oblique.

While Ventura discussed Beckham's potential return prior to Tuesday's game, he wasn't about to discuss the player who could be moved off the active roster for Beckham.

"I'm not getting into that yet," Ventura said. "Anytime a guy comes off the DL, it's always somebody, but we'll figure that out when it gets closer."

• The White Sox entered Tuesday's action with a 7-4 record against the American League Central after going 26-50 within the division in '13 and setting a franchise record for losses within the division.

Alexei Ramirez has hit in 19 of 20 games.

Scott Merkin is a reporter for MLB.com. Read his blog, Merk's Works, and follow him on Twitter @scottmerkin. This story was not subject to the approval of Major League Baseball or its clubs.