Hawkins went on the DL, retroactive to June 10, with nerve irritation in his right elbow.
"Nerves tend to heal slowly, but we don't think this is going to be long-term," trainer Stan Conte said.
After San Francisco beat Detroit 4-0 on Friday night, Hawkins said he was concerned about his ailment.
"If your fingertips were tingling for the first time, you would be concerned, too," Hawkins said.
Hawkins is 0-1 with a 15.75 ERA in five appearances with the Giants since they acquired him on May 28 from the Cubs for right-handers Jerome Williams and David Aardsma and cash. He gave up two runs in his last outing, a 9-7 victory over Kansas City on June 9.
Puffer was 5-2 with a 3.99 ERA in 27 appearances at Triple-A Fresno. He has major-league experience with Houston and San Diego.