Take a step back if it's on the ground and not dead. Shoot it in the head/neck from an angle and you will be fine.
With the older standard lead nose rounds it used to be more common to come up on a deer that was mortally wounded, but not dead yet. They all took a finishing shot, or cut the deers throat, and then claimed their kill. Do what is easier for you. Most guys made another shot because a wounded deer can hurt you seriously if you get too close.
When I shot my first deer it was still alive when I got to it. I had taken out a lung, and it was down for the count. I used the rifle my father had me using, a .44 mag, and fired a close range shot to finish it off.
If you are worried about something that is so unlikey to hurt you, then carry a smaller caliber sidearm.