I've made a number of canes; since my lathe has an 18" bed, I've had to make them in sections that were glued together. I worried a lot about the strength of those joints, and that forced me to do some rather fussy things to address those concerns.
But in this instance, you are simply extended a walking stick by 6-8". Seems to me that a joint near the end of a stick is not likely to experience as much stress as a joint near the middle of a stick, so I think it would be adequate to just turn an extension, and drill a hole that matches the existing tenon, and then glue the two pieces together. I would probably make the extension out of contrasting wood so that it appears to be a decorative 'sock' at the end of the stick; that would create the impression that you intended to do it that way rather than realized that you made the stick too short and had to add an extension.