While Oliver's return resolves the matter of a left-hander for the bullpen, Garland's decision not to accept arbitration leaves the Angels one starter down in their rotation as he enters the open market in pursuit of a long-term deal along with Teixeira and Rodriguez, whose decisions to decline were expected.
As a Type B free agent, Garland will return as compensation a sandwich pick between the first two rounds in the next First-Year Player Draft if he signs with another club.
Because they are Type A free agents, Teixeira and K-Rod would command two picks in compensation -- the team's first-round choice if it's not in the top 15 of the Draft -- and another in the compensation round between the first two rounds if they sign with other teams.
Oliver, 38, is coming off one of his finest seasons. In 54 games covering 72 innings, he was 7-1 with a 2.88 ERA. He also delivered handsomely in 2007, his first season with the Angels, going 3-1 with a 3.78 ERA in 61 games.
Oliver earned $2 million in 2008 and is expected to draw a raise in arbitration. The Angels will submit a bid, Oliver will counter, and they'll either agree on a number or go to an arbitrator.
Garland, 29, had an up and down season, finishing 14-8 with a 4.90 ERA in 32 starts. He pitched 196 2/3 innings, ending a string of four consecutive seasons of at least 200 innings with the White Sox. He was acquired last winter from the White Sox in exchange for shortstop Orlando Cabrera.
The Angels will take one of three courses: pursue a fifth starter in free agency, trade for one or throw it open to competition from a group headed by last year's sixth starter, Dustin Moseley. Kelvim Escobar, recovering from shoulder surgery, isn't expected to be up to speed until sometime around midseason.
With outfielders Garret Anderson and Juan Rivera also in the free-agent marketplace, the Angels could lose as many as five players -- fully 20 percent of the club that won the fourth American League West title in five seasons -- if Teixeira, Rodriguez and Garland also depart.
Signing Teixeira, which won't be easy, remains the club's highest priority as the Winter Meetings begin on Monday.
Lyle Spencer is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.