Lets start with a pair of White Sox relievers since they are going to be fresh on everyone's minds. Keynan Middleton is one of the top tier rental reliever targets that is likely to be available at the trade deadline. He has seen his strikeout rate spike this season and the rest of his Statcast profile looks really good other than his walk rate which is below average.
The biggest issue with acquiring Middleton at the deadline for the Braves is likely to be demand. Most, if not all, contenders are going to be looking to upgrade their bullpens for the stretch run and that could make Middleton a bit pricey for a rental. However, assuming things don't get too crazy on the market, Middleton should be high on the Braves' wish list.
The other Chicago reliever the Braves should keep an eye on is Kendall Graveman. Graveman's peripherals aren't quite as good as Middleton's, but he is generally good at limiting hard contact and he sports a 3.07 ERA in 42 appearances this season. He also has the added bonus of being under contract through the 2024 season.
Acquiring Graveman may be a little trickier. One, he is making $8 million a year through 2024 which could be less than desirable from a payroll perspective. Second, that extra year of team control is going to make him cost more in prospects and there is a chance that the Braves simply don't value him highly enough to make that move. There is also a chance that the White Sox think that they can compete next year (they probably could given how bad the AL Central is) and may prefer to hang on to Graveman for that push.
It isn't too often that we look to the Rockies for pitching help, but Brent Suter has been quite good for them in 2023. He currently sports a 2.81 ERA in 31 appearances for the Rockies this season and that is coming off a pretty successful run with the Brewers the previous few years. He doesn't miss a lot of bats, but he is elite at generating soft contact which has proven to be very useful playing at Coors Field.
Other than not missing a ton of bats, the biggest issue with acquiring Suter at the deadline is that he is currently on the shelf with an oblique injury. Obliques are pretty notorious for being fickle when it comes to healing, but it does sound like he is about to start a rehab assignment. This could be a chance to buy a little low on rental reliever who isn't making a ton of money that could be a nice lefty pickup for the bullpen in the second half. There would be some risk given that he is dinged up, though.