Giants’ Will Smith likely to join bullpen next week

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The addition of left-hander Will Smith to the Giants’ bullpen is close to becoming a reality, finally.

Smith anticipates being available to pitch in a big-league game next Tuesday after he completes his long rehab from Tommy John surgery.

“It’s nice to finally get out there under the lights again and playing baseball,” said Smith, who appeared in his fifth minor-league rehab game Sunday. “It’s definitely fun again.”

Smith would give the Giants a rare combination of two late-game lefties, including Tony Watson, whose ERA in 10 appearances is 0.84.

“You’re getting a proven setup-type left-hander with a lot of experience and really good stuff, so it’s going to make the bullpen even better,” manager Bruce Bochy said.

Acquired from Milwaukee at the 2016 trade deadline, Smith hasn’t pitched for the Giants since Sept. 28, 2016. He had surgery to repair the ulnar collateral ligament in his left elbow in March 2017.

The final three rehab outings will be with Triple-A Sacramento, including back-to-back assignments Wednesday and Thursday. The finale is Sunday.

SCOTTSDALE, AZ - FEBRUARY 20: Will Smith #13 of the San Francisco Giants poses on photo day during MLB Spring Training at Scottsdale Stadium on February 20, 2018 in Scottsdale, Arizona. (Photo by Patrick Smith/Getty Images) less SCOTTSDALE, AZ - FEBRUARY 20: Will Smith #13 of the San Francisco Giants poses on photo day during MLB Spring Training at Scottsdale Stadium on February 20, 2018 in Scottsdale, Arizona. (Photo by Patrick ... more Photo: Patrick Smith / Getty Images Photo: Patrick Smith / Getty Images Image 1 of / 1 Caption Close Giants’ Will Smith likely to join bullpen next week 1 / 1 Back to Gallery

“We’re close, we’re getting there,” Smith said. “The trainers are happy with it, we’re bouncing back fine, we’re recovering fine.”

The Giants’ bullpen ERA entering Monday was 3.35. Hunter Strickland, the closer in Mark Melancon’s absence, picked up his fourth save Monday night and lowered his ERA to 1.80.

Melancon, who had a stem-cell injection in his forearm April 12 in the wake of his September pronator surgery, will play catch Thursday, the start of his throwing program.

The longest at-bat: Brandon Belt didn’t bother watching the replay of Sunday’s 21-pitch at-bat. It lasted 12 minutes and 45 seconds, after all.

“No, I’m not going to watch a 12-minute at-bat,” he said. “I’ve got too much stuff to do.”

Belt said he saw a “quickened version” of the at-bat, major-league baseball’s longest since at least 1988 when official pitch counts were recorded. “It was kind of cool. It still doesn’t make much sense how you see 21 pitches in an at-bat.”

Explaining the smile he flashed in the final moments of the at-bat, Belt revealed an exchange with Angels catcher Martin Maldonado.

“I think I just told him I was going to bunt,” Belt said. “I’m trying to put the ball in play. I hate when other people do this, I might just be putting down a bunt right here.”

John Shea is The San Francisco Chronicle’s national baseball writer.