60 degrees turn? You can get better ones.
What matters is how many ohms range you get from stop to stop and how many ohms your hardware uses. Ohms is resistance. I would start with the pot you currently use and put a meter across one of the outside legs of the pot to the center leg to find out what you got.
The standard plug into the gameport stick runs from 0 to 100k ohms. 1/100th of a turn covers 1k ohms. If I put that in series with a 10k pot then that pot will be 1/10th as sensitive as the 100k. The digitizer measures how much voltage gets through with 0 ohms meaning full and increasing ohms going towards none, 110k total as opposed to 100k is not a big deal and should only affect calibration.
That's why I tell that you can have one pot/slider for coarse adjustment and the other for fine adjustment. If you want 20x fine then for a 100k coarse you would add a 5k fine that would change the total resistance 1/20th as much for the same turn as the coarse knob.
Maybe the better setup would be a coarse long slider and turn pot for fine adjustment.
What your stick uses for values, you will have to find out for yourself. If you're up for it there are hobby controller electronics in the $60 and under range like the BUO-series at leobodnar.com (has 12-bit A/D) and less-accurate (10-bit) but far cheaper DIY MCU's like the Teensy's and Leonardo's (that can use external ADC chips to get more bits resolution but that's 'cheating', hehe) and make your own stick which is not for the faint of heart or mind.
The hard part could be making acceptable knobs, grips, and bases, ie the mechanical bits. Plaster molds and bondo then hours of sanding, filling, etc?
I can code the latter and breadboard them but have no shop space and am lousy at soldering and fine work due to shaky hands. The newer controllers and software (since 200
make it easier than ever but I say that already knowing C since the 80's. Your mileage may vary, learning to code from zip may not be worth what you get. There's a lot of help available from Leo for his products (already coded) and on the Arduino forum for those.
In the last two years I have recovered/relearned a lot of my old "skillz" playing with Arduino. I have even made leds that self-adjust with changes to ambient light (LOL Fehler, how much resistor? MINE changes itself!).