I have had more than a plenty of occasions to carry a shorty felling axe aloft, when topping trees for wildlife habitat prescriptions.
My first attempt was to drill a 3/8" diameter hole in the handle near the end and tie a 3 or 4 inch loop of paracord through it. Then clip to a biner on my saddle.
That sucked the hairy moose lips. The damn thing swung all over hell and gone, like to cut my harness, or leg, in two. A sheath kept the cutting danger at bay, but the swinging weight and the snagging still sucked.
So I girth hitched a short sling under and over the head, and hung it from there. Hardly any swing, all the weight was up close to my harness. That was the best solution for me.
Jed, I suspect, experiences something similar, clipping into that loop welded to the axe head. Not great, but better than all the other options.