Should I utilize a different approach to sewing a ~round hole versus a thin cut?

I would cut a couple darts out of each end of the hole then sew.
Any small puckers will come out in the blocking.

Boco, what is a dart?

A V shape cut out and re sewn.

A V shape cut out and re sewn.

What do you mean by puckers & blocking?

Is that a tanned hide in the pic?

Yes.

I had every hole sewed prior to tanning, but when I got it back none of the thread was left in tact. I think my sewing helped hold everything together during the tanning process, but I need to re-sew a few spots. This was my 1st otter put-up. While I'm satisfied with the outcome from the tannery, I didn't do a perfect job prepping it & therefore I have a few blemishes to fix. It's a wall hanger, not to be made into a hat or anything.

Thanks

OK puckers are the little wrinkles left after stitching.
Blocking is wetting the leather and stretching and pinning it to take out excessive stretch left after tanning.

Here is a blocked out coon.

Fold the hole in half along the length of the hole, skin side out. Nip off the two little points that look like the bow and stern of a canoe to make the hole straighter. Sew up with nice, tight stitches.

Sewing before tanning is a waste.They are going to thin that flesh side with a powered round knife and it will take out any stitches.What is being described,is correct.You want to create two little points at the end of that oval hole,like a football shape.I use a SHARP clovers needle and Berkely Fire Line,8-10 lb and fine stitches.

There are a whole bunch of different techniques that furriers use to repair/prepare pelts.
The basic one is elongating the hole like a football shape as was stated.There comes a point where because of size or proximity of other smaller holes that method wastes too much sq inches of useable fur.That is when other techniques are used.

There are a whole bunch of different techniques that furriers use to repair/prepare pelts.
The basic one is elongating the hole like a football shape as was stated.There comes a point where because of size or proximity of other smaller holes that method wastes too much sq inches of useable fur,or alters the character of the skin.That is when other techniques are used.
Sewing before tanning is a benefit as it will mean the holes will be smaller and easier to repair when the stitches come out during the final buffing.This is especially true if you are tanning your own skins.
The tannery I use will repair holes before returning them if they open up during the process.

This is my belief too.