MINNEAPOLIS -- White Sox manager Ozzie Guillen was asked on Wednesday if general manager Ken Williams came to him with the offer of one addition needed to make this team better as the July 31 non-waiver Trade Deadline approaches, what position would he request? Guillen didn't take long to point at improvement from his own players as his lone suggestion.

"I need [Alex] Rios and [Adam] Dunn to start hitting," said Guillen. "To be honest with you, I'm not going to say we need this or that. When we set this ballclub up, spent a lot of money on this ballclub, [we] assume they're going to do what they're supposed to do.

"When these guys get hot for a little while, then we'll go for it. Right now, we have a six-man rotation. We can't put another guy out there. We're fine. Everything has worked out pretty good. Our bench is better. Hopefully, these guys start hitting and help the ballclub."

Teahen adjusting to sporadic playing time

MINNEAPOLIS -- Mark Teahen certainly wants to be a part of the 2011 White Sox, a very good team which he believes has the chance to do special things.

To say Teahen also is satisfied with his part-time role, limiting the left-handed hitter to two starts in June and 59 injury-influenced at-bats all season, would not be an accurate reflection of the competitor that is the 29-year-old.

"If I said I'm happy not playing every day, that would be completely untrue," said Teahen, speaking before Wednesday's contest with the Twins at Target Field. "Obviously, I want to be in there showing what I can do on a daily basis.

"But they are asking me to do this role right now and it doesn't help me at all to complain about it. I'm trying to make the most of the chances I get when I do get in there.

"I remember reading Juan Pierre saying in Los Angeles after they signed a couple of outfielders, 'If you are looking for a guy happy to sit here on the bench and collect a paycheck, you got the wrong guy,'" Teahen said. "I echo that. Obviously, I play this game because I want to play and help the team win."

Don't think for a second these comments represent sour grapes on Teahen's part. More like disappointment from a productive player over five seasons with the Royals, who has battled injuries and struggles since joining the White Sox via trade on Nov. 6, 2009.

Even Teahen's super-sub role has kind of been transferred to Brent Lillibridge, as Teahen missed time due to a strained right oblique. In fact, a smiling Teahen couldn't exactly define his present set of responsibilities.

"Role to be defined, I guess," Teahen said. "I know I'll get a few games at third and a couple at first or some outfield. At this point, it's more or less giving a day off when you can. Like Lillibridge, he did a great job and got a few more opportunities. I just have to continue to work hard and make sure to make the most of those opportunities and see if they come more rapidly or often.

"Right now, I kind of look at where I'm at and think it's crazy I'm in this spot. I know how injuries go and I know when you are in a deep organization and other guys come in and do a good job. I have to be patient or whatever else, but it's aggravating to see other teams come in and know I'd be starting at two or three positions for some of those teams.

"It's weird at this point in my career to not be getting in every day," Teahen said. "It's a huge adjustment because I've always been preparing to be in there every day and it's much easier to get in a rhythm playing every day. But it is what it is. It doesn't help to whine about it. Just make the most of it."

Peavy's rehab performance to determine return

MINNEAPOLIS -- Jake Peavy's Wednesday flight to Charlotte was slightly delayed, but he will be with the Triple-A Knights for his rehab start on Thursday night against Syracuse.

Peavy was hoping to make one Minor League start, covering six innings or 90 pitches, to test his right groin strain in live game action and then return to the White Sox rotation sometime next week. But manager Ozzie Guillen explained that Peavy could need more than one rehab start.

"That's up to how he feels, how many pitches he throws and how we look at him and how hard he went through it," said Guillen. "There will be a lot of things that they watch to see how many days he'll be down there."

Guillen, general manager Ken Williams and pitching coach Don Cooper all talked again on Wednesday about what to do with the rotation once Peavy is healthy. Once again, the decision was to let Peavy get healthy and then figure out the arrangement.

"I don't have anything specific now about what we should do," Guillen said.

Ozzie pays tribute to Venezuelan umpire

MINNEAPOLIS -- Wednesday's White Sox lineup card was brought out by manager Ozzie Guillen, instead of third-base coach Jeff Cox. Guillen made the move in order to pay tribute to Manny Gonzalez, the second-base umpire and the first Venezuelan-born umpire.

Guillen chatted with the crew and then took a picture with Gonzalez at home plate.

Third to first

• Reliever Tony Pena also joined Peavy on a Minor League rehab assignment to Triple-A Charlotte. The reliever has been out since May 28 with right elbow tendinitis.

• The White Sox were rained out of batting practice again on Wednesday, meaning Thursday will be the first day they have hit since Saturday at home.

• Juan Pierre's 19 hits are the most for any opposing player at Target Field.

• Gordon Beckham's sixth-inning double off Carl Pavano in Wednesday's 4-1 loss was his first double in 28 games.