What's the best way to make a potentiometer shaft longer?

Not a lot, and that's where the problem arises.
An extra 5-6mm should do it, any "sleeve" I put over the existing shaft is likely to be in the way..
It is why I asked, as couplers won't suit a short extension. Also, should supply a bit more detail so we may help devise an effective strategy. Things like size type of shaft. Original uncut D or already cut short?

I would slit the end of the shaft using a dremel cutoff wheel. Then shape and fit a flat and hard bit of steel into the slit with length required for extension. Then cover the flat hanging out of the slit with an epoxy putty. You can hand sand that back into the shaft shape after cure
 
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I'm pretty sure any kind of glue by itself will not make a permanent repair.

Assuming it's a round (smooth) shaft, I would look at cutting two flats on either side of the pot shaft, leaving a tab in the middle that's about 1/4 to 1/3 the width of the shaft, perhaps 1/4" long (3–4 mm). Then find a piece of aluminum rod the same diameter as the shaft, and cut a groove the same width and depth to fit the modified shaft, so they form a good mating pair. Tape those securely together and drill a small (1/16"/ 2–3 mm) hole through both parts, the diameter of which should match some aluminum/copper/brass wire or rod you have. Cut the pin to be just slightly longer than the shaft diameter, then push the pin through and gently pound the ends down so they widen a little and lock the pin in place. File the pin ends down to the rod surface. Test-fit beforehand, and if it's a bit wobbly, a little epoxy would stabilize it. File a new flat on the extension if needed.

Aluminum rod can be found at hobby or hardware stores, or of course online. An X-acto micro saw and a file should do the job. Thin-walled brass tubing is available from the same sources, and could be slid over the repair if your fit-up is less than perfect. If the pot has a splined shaft, that's impossible to create from scratch, so you might see of there is another matching spline on a pot in your junk drawer that could be cut off and used for an extension.

I know this sounds like a lot of fussy work, but this will create a really permanent repair you'll never have to think about again. Or, you can just buy a new pot with the right shaft length.
 
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