CHICAGO — The non-waiver trading deadline may have passed last week, but it seems clear that the Red Sox aren’t done dealing.

The Sox yesterday claimed San Diego Padres outfielder Brian Giles and were attempting to make a deal with the Padres for the 37-year-old lefty hitter.

Giles was among a number of players who cleared trade waivers this month in the aftermath of last week’s non-waiver trade deadline. Players must first clear waivers before being dealt after July 31.

Boston’s interest in Giles may be twofold. The Sox could use some depth off the bench and Giles, though no longer much of a power hitter — he has just six homers in 389 at-bats — is still productive at the plate, sporting a respectable OPS of .815.

Second, the Sox could keep Giles away from the Tampa Bay Rays.

The Rays were interested in dealing for outfielder Jason Bay before the Sox obtained him in the three-way trade that sent Manny Ramirez to the Los Angeles Dodgers.

Ironically, Giles was once traded for Bay, whom the Sox obtained last week.

In a waiver deal in 2003, Giles went from the Pittsburgh Pirates for Bay, Oliver Perez and a minor-leaguer.

Looking for depth

Beyond a bat off the bench, the Red Sox are also said to be searching for some middle-infield depth with an eye toward expanded rosters in September.

Shortstop Julio Lugo is sidelined until at least the end of August, and while Alex Cora has played well of late, filling it at both short and second, he has lost some range.

The team would be interested in a defense-minded infielder who could come off the bench in late innings and provide depth at the position.

One possibility might be former Providence College standout John McDonald. McDonald is hitting just .185 for the Toronto Blue Jays, but is one of the better defensive infielders in the game.

One caveat: McDonald is under contract next season for $1.8 million, an obligation the Sox might not wish to take on. Then again, with Cora a free agent, McDonald could simply take over the role of utility infielder next season and replace Cora on the roster.

Smith demoted

As expected, the Sox returned pitcher Chris Smith to Pawtucket yesterday. Though the team didn’t announce a corresponding roster move, it’s certain to be the recall of David Aardsma from the PawSox.

Aardsma had been on a rehab assignment with Pawtucket after suffering a groin pull shortly after the All-Star break. He made two appearances for the PawSox earlier this week, last appearing Tuesday night in Scranton/Wilkes Barre.

Smith, meanwhile, pitched two times for the Sox during this stay with the parent club, tossing two scoreless innings. Overall, in two stints this season, Smith is 0-0 with a 2.16 ERA in five appearances.

Two draftees sign

The Sox announced the signing of two picks from June’s draft — sixth-round selection Ryan Lavarnway, a catcher from Yale, and 36th-round pick Richard Wasielewski, a high school lefty from St. Simons, Ga.

Lavarnway, who visited with the Sox when they were in Anaheim last month, was a semifinalist for the Johnny Bench award — given annually to the best college catcher in the nation — and the Golden Spikes Award, awarded each year to the nation’s top player.

Lavarnway will start his pro career at Lowell of the New York-Penn League while Wasielewski was sent to the Gulf Coast Red Sox in Fort Myers.

The Sox have until Aug. 15 to come to terms with their remaining players drafted, including Portsmouth standout Ryan Westmoreland.

Bay’s fast start

Bay has hit safely in each of the six games he’s played since joining the Red Sox last Friday.

According to research by the Elias Sports Bureau, the last midseason acquisition to begin his Red Sox career with a hitting streak of at least six games was third baseman George Kell, who hit in 10 straight after coming over from Detroit in June of 1952.

Bay is hitting .423 (11-for-26) in a Sox uniform, with six RBI. He also has scored 10 runs, the most of any player this month.

It’s about time

Incredibly, though the season is 115 games old, tonight represents the first meeting of the year between the Red Sox and White Sox.

The teams will play seven games between tonight and the end of the month.

The Red Sox have won the last seven meetings, including a four-game thumping at U.S. Cellular Field last August, outscoring them, 46-7, and scoring at least 10 runs in every game.

The series marked just the third time in franchise history that the Red Sox had managed to score double figures in runs in four consecutive games. They also did it in 1912 and 1950.

Jon Lester, who will pitch the opener of the series tonight, has won his last seven decisions. That’s the longest winning streak by a Red Sox lefty since Bruce Hurst had a seven-game run in 1988.

smcadam@projo.com