The Yankees announced Wednesday that Nady's second Tommy John operation will be done July 8. Nady also had the surgery in 2001.
"I'm extremely disappointed for him because we thought we were going to get him back this month," manager Joe Girardi said. "He'll be back. Your heart goes out to him because it's a lot to deal with.
The injury could also cost Nady a considerable amount of money because he can become a free agent after the season.
Nady was placed on the disabled list on April 16, a retroactive move two days after he felt a sharp pain in his right elbow while making a throw at Tampa Bay.
On Tuesday, Nady saw Dr. Lewis Yocum and the news the Yankees received Wednesday was expected -- Nady needs another operation.
The Yankees sent two minor leaguers to the Pirates: right-hander Casey Erickson and outfielder Eric Fryer. New York also received cash from the Pirates in the deal.
Hinske's contract pays him $1.5 million this season.
After Wednesday's 4-2 win over the Seattle Mariners, the Yankees announced that catcher Jose Molina, who has been out with a strained left quadriceps muscle, will make a rehab appearance for Scranton/Wilkes-Barre on Thursday at Pawtucket.
The 2002 AL Rookie of the Year with Toronto joined his fourth AL East team. He played for Tampa Bay last season and Boston the year before, playing in two consecutive World Series.
Pena was batting .246 with four RBIs in 46 games, and the Yankees plan to have the switch-hitting infielder play different positions to improve the his versatility.
"He's going to play some short and some second and some outfield and do a lot of different things down there," manager Joe Girardi said. "He'll play some center and we'll start there and we'll probably move him around a little bit.
"He's an interesting guy. He's a switch-hitter that hit very well left-handed and in the past he's said he's been more comfortable right-handed. He didn't get a whole bunch of right-handed at-bats. Only 21 at-bats in the month of June. We feel that he needs more at-bats."
The addition of Hinske gives Girardi a patient hitter whom he can write into the lineup at any of the four corner spots -- first base, third base, left field and right field.
He will be a bench player in New York, a role he filled on the 2007 Red Sox. He began moving around the field after three seasons as a regular starter at third in Toronto.
His flight was delayed by weather on Tuesday night, but now he's back in familiar surroundings.
"It's good to be on a team that has a chance to gain first place or a spot in the wild card," Hinske said.
A year after he was on the championship-winning Red Sox, he made the last out of the World Series, striking out as a pinch-hitter against Brad Lidge as the Philadelphia Phillies finished off the Rays in five games last October.
Although he has taken to the utility role well, Hinske said he didn't feel as if he fit in Pittsburgh, where he frequently was called upon to pinch-hit.
"I missed the way the game is played here," Hinske said. "I missed the American League."