First, he comes across what looks like a dead deer. He finds that the deer is an old and injured buck. The deer attempted to get up, and Aaron said, seeing it stand gave me hope that it would be OK... I wanted it to recover." From this, one can infer that Mr. Hitchins cared about the fairness and health of his game and did not want to shoot the deer.
Next, he gave up his entire week for the dignity of the "old warrior". By wrapping his tag around the deer's antler, he would end his week after only a few hours, and that's exactly what he did. The narrative states