So, I decided to take an old umbrella and modify it so that one item could do the job of both.
Step 1: Materials Required
I went through my scrap pile and found some 1/2 inch diameter copper tubing and a wood dowel that fit inside of it. The tubing was a perfect diameter to slip over the end of the umbrella.
Since I had no cane tips that would fit a 1/2 inch diameter shaft, I carved a rubber plug from a piece of 1/2 inch thick dense rubber for a cane tip. When this rubber plug eventually gets worn down, I'll replace it with a regular cane tip.
Step 2: Assemble and Install the Extension
I cut the dowel so that it would contact the end of the umbrella tip, and leave about 1/4 inch of clearance between it and the end of the copper tubing (1st photo).
I then coated the rubber tip with epoxy, and forced it into the end of the tubing until it contacted the dowel (2nd photo).
The idea is that the weight exerted on the rubber tip is transferred to the dowel, then to the shaft of the umbrella. The copper tubing is simply there to keep everything aligned.
Step 3: Completion
I have found this to work very well, both as an umbrella, and occasionally as a cane. Now instead of carrying both in my truck, I do just fine with one which can serve two purposes.