What about wrapping some clear fishing line and then using the clear 3M hooks?

Please post a couple of photos of your fireplace and room -- there might be more suggestions ....

... perhaps consider placing some old leatherbound books on your mantel to "hold" the walking cane -- and then add a framed photo or two of the gentlemen .... don't forget to stash names/dates etc. on the back of the framed photo/s -- always good for future generations! :)

.. Or perhaps consider creating a family gallery wall -- and place the walking cane on a wall shelf amongst the photos (and other wall shelves with other family heirlooms) -- with a family tree ......

If it is going to lay horizontally, then how about two small cradles sitting on the mantel to rest it on, one taller than other other?

They sell golf club display cases . . .

Two clear fishing line strands ... one holds the cane at the crook or the knob, the other is at the tip.

The other end of the line is in a small nail way up in the molding.

We used L-shaped screws and screwed them into the wall at the appropriate depth to fit each palica or shillelagh.

Look at some of the less tacky gun hooks. Nothing with bullets as part of the design. :x They are padded and slip resistant and you can always use fishline or something for extra security.

Like these, but in a finish that works with your decor?

Here is a link that might be useful: brass hooks

AnnieDN, your arrangement is the one I thought of even before scrolling down. I've always liked the way the angle of the canes repeats the angle of the rail on the other side of the doorway.

Have it properly framed in a shadowbox, in a vertical orientation. Then place it on a gallery wwall of similarly framed family pictures and mementos.

The way I hang my yard tools, also with tapered handle, is I hammer in two nails, then hang the rake, or pitchfork, or whatever, by putting it between the two nails where it's narrow enough to fit, and then letting go and it is held in place by the handle getting too big. YOu could do the same with the cane.

Thank you all, I never would have thought of some of those suggestions. I'm sure one of them will work.
The fireplace mantel and surround are painted wood. Right under the mantel is a blank expanse where the cane would be anchored by the vintage mirror and paintings above. I want to angle it slightly, like the Nike "swoosh".
I can't find a single picture of the fireplace. It looks like a lot of my pictures were lost when I got my new laptop. I'll have to take some more.

Thanks, mama goose!