I dislocated my shoulder some time ago. I went to physical therapy for rehab, but now I have been discharged. I notice that when my arm is at my side and my elbow is bent, I can move my hand away from my body much further on the injured side compared to the other side. Why is this?

This may actually be a sign of injury to the structures around the front of your shoulder. This type of excess motion can persist even after rehabilitation. It may indicate the need for surgery, especially if the shoulder continues to dislocate or cause extra pain. You should see your doctor for a follow-up evaluation.