Angels report: Joe Smith close to minor league rehab stint

Joe Smith threw live batting practice Thursday, his second mound session since a left hamstring strain sent him to the disabled list June 7. Barring a setback, the setup man could begin a minor league rehabilitation stint as early as Sunday.

Smith hopes to be activated for the July 1 start of a 10-game trip to Boston, Tampa Bay and Baltimore. He has not come to grips with the possibility that his return could amount to a month-long showcase for contenders looking to add relief help before the Aug. 1 non-waiver trade deadline.

With the Angels trailing Texas by 15 1/2 games entering Thursday and Smith in the final year of a three-year, $15.75-million deal, the sidearm-throwing right-hander could be the most likely Angels player to be traded.

“That would be tough to swallow,” Smith, 32, said. “I came here because I thought we had a legitimate chance to win. In 2014, we rolled [to 98 wins and a division title] and were swept in the playoffs. In 2015 we grinded the whole way until the last day of the season.

“I didn’t sign here to play 2 1/2 years and leave. I signed because of guys like Mike Trout, Kole Calhoun, Garrett Richards, Jered Weaver and Albert Pujols. We have a damn good team. You want to play your whole contract out. You want to win.”

Smith hurt his leg May 19 against the Dodgers, but he didn’t tell the team because closer Huston Street was on the DL, and he felt he needed to play hurt. When the discomfort worsened over his next five appearances, Smith had no choice but to speak up.

As long as it has taken his leg to heal, it has been more painful for Smith to watch the Angels lose 13 of 20 games in June and drop 8 1/2 games in the standings.

“It’s terrible to be sitting there while the team is struggling,” Smith said. “You especially want to play when you’re losing because you feel like you can bring something to the table. Sitting like a fan in the dugout, watching these guys grind, it’s a hopeless, lost feeling, because I should be grinding with these guys.”

Johnny off the spot

Second baseman Johnny Giavotella made a spectacular play in the eighth inning of Tuesday night’s 3-2 loss at Houston, diving to his right to stop Jason Castro’s grounder and, with his glove hand, shoveling a flip to shortstop Andrelton Simmons to start a double play and preserve a 2-1 lead.

In the ninth inning, Gregorio Petit replaced Giavotella, a strategy Manager Mike Scioscia will continue to employ despite Giavotella’s clear improvement on defense this season.

“Johnny has worked very hard at it, and he’s made strides defensively,” Scioscia said. “But there are things Petit can do that are real special on the defensive end. It’s not a slight against Johnny. He’s playing much better, no doubt. But if we have a chance to upgrade, we’re going to take advantage of it.”

Short hops

Catcher Geovany Soto, out since May 18 because of right knee surgery, has been hitting, running, throwing and catching bullpen sessions and hopes to begin a minor league stint late next week. …

Tyler Skaggs (shoulder discomfort) threw 2 2/3 innings and 45 pitches in an extended spring training game in Arizona on Wednesday but will need several more outings before rejoining a minor league affiliate….

Nick Tropeano, nearing the end of his rehabilitation from right shoulder tightness, threw five innings for Class-A Inland Empire Thursday night, allowing two earned runs and seven hits, striking out seven and walking none. Of his 71 pitches, 54 were strikes.

mike.digiovanna@latimes.com

Twitter: @MikeDiGiovanna