Not according to the sequence of events recounted by John Taylor, which you yourself quoted:
I shall never forget the deep feeling of sympathy and regard manifested in the countenance of Brother Joseph as he drew nigh to Hyrum, and, leaning over him, exclaimed, 'Oh! my poor, dear brother Hyrum!' He, however, instantly arose, and with a firm, quick step, and a determined expression of countenance, approached the door, and pulling the six-shooter left by Brother Wheelock from his pocket, opened the door slightly, and snapped the pistol six successive times; only three of the barrels, however, were discharged. . . .
Hyrum Smith was shot before Joseph Smith returned fire with his six-shooter. Moreover, Joseph "opened the door slightly" and fired at the attackers outside the room; Hyrum was on the floor inside the room.
Well, this indeed may be the sequence. But just because you tell a sequence doesn't mean ALL of the details were included -- especially if Taylor didn't know every detail!
There was a reason Hyrum was at the door; and it's likely because those from the mob had already opened fire (I doubt the first shot is the one that shot Hyrum in the face). So if Hyrum is at the door, the shooting has already begun from people just outside the door. And if Joseph is trying to shoot one through the door, he may have missed & shot Hyrum in the back right around the exact same time as Hyrum was shot by a member of the mob through the door!
It doesn't change who killed Hyrum. But it's really the only explanation for a back wound of a man who is laying on the floor, back-down, since he was killed.
If John Taylor is to be believed. He would have reason to “correct” the story in order to let the prophet ‘save face.’