Oliver has spent the past two years with the Rangers and still lives with his wife, Melissa, and their two sons in Northeast Tarrant County. Ten of his 18 years have been spent with the Rangers, the team that originally drafted him in the third round of the 1988 First-Year Player Draft.
He wouldn't mind ending his career with the Rangers.
"I don't see why not," Oliver said. "Why wouldn't I? I'm going to leave the door open."
The Rangers had two left-handed relievers in the playoffs. Michael Gonzalez was the other, and he is also a free agent. Michael Kirkman is still on the roster, but left-handed relief will be an area that the Rangers need to address this winter.
Oliver was 5-5 with a 2.29 ERA in 61 games for the Rangers this year as their primary left-handed setup reliever. Starter or reliever, it was his lowest ERA for one season in his career. Opponents had a .280 on-base percentage, which was also the lowest of his career.
He has now pitched in the playoffs for six straight seasons, including the Mets in 2006 and the Angels in 2007-2009. His 30 postseason appearances are tied for 17th all time.
He was on the mound in Game 6 of the World Series last Thursday in St. Louis. He took over in the 10th inning with a 9-7 lead but gave up two singles to left-handed hitters, and the Cardinals rallied to tie the game. They won it in the 11th and then won Game 7 on Friday night.
"There are so many scenarios that go through your mind about what could have happened," Oliver said. "The bottom line is we lost. We got beat. The scenarios give you a headache when you try to think about it. It stings. It's going to sting for a while for a lot of guys, not just myself. That was a tough one to take.
"You try not to think about it, but then you wake up in the middle of the night and you can't help but think about it. As the days go by, it gets better. I woke up on Saturday and the sun had come out. But it's weird how it ended so abruptly."
He does not want his long and successful career to end abruptly like this.
T.R. Sullivan is a reporter for MLB.com. Read his blog, Postcards from Elysian Fields, and follow him on Twitter @Sullivan_Ranger. This story was not subject to the approval of Major League Baseball or its clubs.