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Basically replace it with an old school friction shifter.
Thanks, it wouldn’t have crossed my mind to do that!The SA and SACHS/SRAM 3-speed shifters, are properly indexed for the 3-speed SA hub. Purely friction shifters can be used, according to the Sheldon Brown technical website, but then you have to memorise the orientation of the shifter when selecting the middle gear --- apparently, it was when it pointed at his navel...
Switch to a lever-type shifter; fitted in the correct place on the handlebar the action is ergonomic. SA manufacture a variety of them. I have been using the same SACHS/SRAM Torpedo shifter since 2002, which works fine with SA 3-speed hubs. A replica of the SACHS shifter is available from SJS Cycles.