I'm in agreement with John Smith on this - the simplest solution might be one of those slip-on rubber feet that are sold in sets of four in hardware stores. There is usually a selection of inside diameters to choose from. If I understand correctly, the diameter at the top of the existing metal ferrule is 19mm, so that size or perhaps slightly smaller might be right.
In the conventional application, those rubber feet slip on and are held in place by friction. The ones I'm familiar with have parallel sides on the inside (John's sketch shows tapered sides), but there are ridges inside that help hold them in place. The fact that you would be slipping a foot over a tapered shaft suggests that you might want to add some adhesive to fill the void that would be left inside. Epoxy would work well, but a polyurethane construction adhesive is another possibility.
If you are concerned about the possibility that the attachment might not be secure and the foot could get lost, you could also insert a wood screw through the rubber and into the shaft - a single screw with a washer to secure it against the side of the rubber foot would prevent the foot from falling off and would not detract from the appearance.
There is a slightly more complicated approach that you might consider if you have the skills (or have a friend with the skills). You mentioned that the length of the cane is correct for you, so I presume that you would not want to shorten the cane by cutting off the metal ferule. However,it would not be difficult to cut off the ferule, and then add a small length of wood dowel to restore the cane to the correct length. I would drill an axial hole through the dowel and into the shaft of the cane, and then glue a wooden dowel through the extension and into the shaft for strength. The result would be a uniform diameter at the bottom of the shaft that would more readily accept a rubber foot.
