The Chicago White Sox are still tearing apart their bullpen in advance of Monday's non-waiver trade deadline, trading veteran left-hander Dan Jennings to the Tampa Bay Rays for minor-leaguer Casey Gillaspie on Thursday.

Jennings joins David Robertson, Tommy Kahnle and Anthony Swarzak as veteran relievers traded by the White Sox in the past week. The club could still look to shed others like Tyler Clippard and Jake Petricka before the deadline.

The 30-year-old Jennings has a 3.45 ERA with 38 strikeouts in 44 1/3 innings this season, holding left-handed hitters to a .169/.286/.211 batting line. The Rays have been looking for a reliable lefty reliever since the offseason. As an added bonus, Jennings will remain under team control through 2019 as an arbitration-eligible player.

Gillaspie, 24, was the 20th overall pick in the 2014 draft. He is hitting .227/.296/.357 with nine home runs in 95 Triple-A games this season. MLB.com ranked Gillaspie as the 10th-best prospect in Tampa Bay's farm system before the trade, saying he "shows the makings of becoming an everyday contributor at the highest level."

The White Sox are undergoing a massive rebuild. The Rays, meanwhile, come into Thursday with a 53-49 record. They're 2 1/2 games back in the AL East and one game back of the second wild-card spot.