If you're out stalking, presumably with a rifle, why do you need a pistol for HD? There's a perfectly capable piece of equipment in your hands already?
2 situations where a rifle is not ideal.
1) when you are using a dog or dogs to bay a wounded animal. More the case with boar than deer, but a rifle is too powerful for close range situations and real dangers from bullet pass through etc. A pistol bullet has much less energy, but sufficient to penetrate into the vitals without having to worry to much about over penetration.
2) when you are in the thick bush crawling after a wounded deer a scoped rifle is kind of unwealdy, a pistol much less so.
But yes a rifle is in hand and probably best tool for the job. Indeed the best tool is time. After any shot wait a good ten or fifteen minutes, especially if the animal has gone out of sight. If in doubt leave it an hour or two.
Most animals when wounded will quickly seek cover, lie down and die peacefully. Follow up too soon they get really stressed, full of adrenaline and then take off.
And often best stay where you are as you should have a good rest etc, or move sideways so that you can see where the animal may have fallen etc.
I have had a few occasions where follow up shot has been necessary. One was a hind that I shot with a soft RWS 243 bullet. Range was a bit further and a bit windier than I thought. Bullet hit the shoulder and the hind went straight down. Stalker I was with was in a hurry so we went straight up to it. It was lying looking dead so put rifle down etc and started getting ready to gralloch. She regained consciousness, struggled to her feet and set off round the hill out of sight on three legs. We followed her for what seemed an age, but probably no more than 20 minutes and eventually got into a position where I put a bullet in back of her head as she was standing and started swaying.
On examination, the first bullet had failed to adequately penetrate and had done limited damage heart and lungs. There was plenty of blood in the cavity.
I suspect if we had just stayed where we were for 20min she would have died where she first fell.