I noticed the button is connected to one of the ADC I/O pins on the RP 2040. This gave me a wild idea, what if you replaced the single action tactile push button with one of those tiny X-Y D-pad-like discrete joysticks, using a resistive divider to differentiate between the different presses.
Alternatively, how about one of those pots (real potentiometer, not an optical encoder) with a push button action? That would also go with the ADC input. But that seems lame, because usually you’d want such a knob to be free spinning multi-rotation. Not sure offhand if those analog pins can be a counter — and certainly a single pin can’t work as a quadrature decoder to pick up direction of rotation.
So the knob ideas are probably not any good. But the discrete joystick might have merit. Could potentially even use the same existing button — I’ve seen some really tiny versions of this type of switch. Now this also assumes you have room on the PCB for a presumably larger footprint and the handful of resistors.