The team announced that bench coach Kirk Gibson has agreed to a two-year extension and bullpen coach Glenn Sherlock had also been offered an extension.
Pitching coach Mel Stottlemyre Jr. and hitting coach Jack Howell, who were under contract, will also return.
That some changes were made was expected given that Arizona manager A.J. Hinch had inherited the current staff except for Stottlemyre and Howell, who were both promoted from within the organization when Hinch was hired to take over for Bob Melvin on May 8.
"They are changes that I felt and we felt as an organization were important moving forward," Hinch said. "They were tremendous workers and did a lot of good for this organization and it shouldn't be looked upon as a negative. I wanted to have an opportunity to make a change and move forward with some different voices in different roles."
Hinch informed Hale and Bundy of his decision Sunday night when the team returned from its season-ending series in Chicago.
Hale had served as third-base coach for the past three seasons, and prior to that had served as a manager in the organization for six years.
Bundy was hired by Melvin last offseason after spending two seasons managing the Dodgers' Triple-A club. He worked in the Arizona organization from 2003-06, including a stint as bench coach in the big leagues in 2004.
"I appreciate the opportunity," said Bundy, who will undergo surgery Tuesday to have a benign tumor removed from his left earlobe. "It's a tough situation. We didn't play as well as expected, and when that happens, there are ramifications."
The D-backs are considering former All-Star third baseman Matt Williams for one of the openings, FoxSports.com, citing a Major League source, reported late Monday.
Williams, who played for the D-backs from 1998-2003, is a minority owner of the D-backs and a part-time color analyst on the team's television broadcasts.
Steve Gilbert is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.