The Rangers traded for right-handed setup men Mike Adams and Koji Uehara last July. They signed free-agent closer Joe Nathan to a two-year, $14.5 million contract last week.

And they are still trying to upgrade their bullpen.

Not only did the Rangers make a strong push for free-agent righty Jonathan Broxton, who signed a one-year, $4 million deal with the Royals, but they also are one of the clubs trying to trade for Athletics closer Andrew Bailey, major-league sources say.

The Rangers, who also pursued Bailey before the non-waiver trade deadline, intend to build the strongest relief corps possible after producing a 7.43 bullpen ERA in their seven-game loss to the Cardinals in the World Series.

“They are going to get one more guy at the back end of their bullpen, I can promise you that,” said one source who has been in contact with the Rangers.

The addition of Nathan enabled the Rangers to move their previous closer, right-hander Neftali Feliz, into the starting rotation. Broxton likely would have pitched in the seventh inning, with Adams continuing to work the eighth.

A trade for Bailey would give the Rangers a late-inning tandem of Adams, Bailey and Nathan — one of the strongest in baseball. A lefty such as the White Sox’s Matt Thornton might be an even better fit than Bailey. And it’s still possible that the team will re-sign C.J. Wilson or replace him with another free-agent lefty, Mark Buerhle.

Uehara could remain part of the bullpen mix, or the Rangers could trade him. Teams are calling the Rangers about Uehara, sources say, but club officials have been reluctant to move him.

— Ken Rosenthal