The 32-year-old central defender suffered the injury in the 26th minute, when he was replaced by Jack Deaman.
Taylor, a summer signing from Bradford City, missed the second half of last season with a foot ligament injury. He was making his 15th appearance for the Robins.
Senior sports therapist Ian Weston said: "As he was going into a sprint, Matt felt a pop in his hamstring.
"He has a palpable mid-belly tear in the muscle which will rule him out for six weeks."