Tada! ("before" pic here)
After stripping off the old fabric, I gave the chair a light sanding, followed by a couple coats of Rust-Oleum's paint + primer in gloss white. I finished off the paint job with a clear protective coating. Note: this was the first and last time I'll use a paint+primer - it offered zero stain blocking, and there was a LOT of over-spray. On a positive note, the paint really had a glossy finish.
Although I purchased upholstery buttons, I ended up re-using the original buttons because they had brads on the back - which meant zero sewing (yay!). I simply cut out tiny circles of fabric, and then glued the fabric onto each button.
Another shot of my lovely refurbished cane chair from a slightly different angle: