Sling stud filler screw?

One of the best ways to hide the hole is to plug it with wood from the rifle itself. Take out the sling and measure how big the previous owner made the hole. It doesn't look like the previous owner put a chamfer hole for the stud to rest on (hence the o-ring). So the plug may have to be an unusual size. Once you know how big of a plug you need, take a plug cutter drill bit and harvest some wood from under the butt plate to plug the hole. Here is a small set of plug bits.

https://www.lowes.com/pd/Milescraft-3-Piece-Plug-Cutter-Set/50192299

After you plug it, you can add stain and finishes to match the rest of the rifle. Wood under a butt stock usually isn't finished like the rest of the rifle is so you will have to do that yourself to hide it as much as possible.

Tallinar said:

I suppose there’s the out of sight, out of mind approach…
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That will work fine as a temporary fix. I only say that because leather will eventually cause the finish underneath to deteriorate over an extended period of time. For instance if you are in the field with it and the rifle inadvertently gets wet. We all know that life happens. Ask anyone who owns a rifle that has been in a saddle scabbard. You have nothing but time on your side to decide which permanent fix will best suit your needs. I think that you have some solid ideas presented to you for a permanent fix. It's all up to you.

For now just remove that white spacer. It will look 100% better. Might even solve your problem.