On a FIRST robot, we needed something similar. There's "full reverse", "full forward" and "I need fine control", and the original mapping of joystick to wheel motors was linear. It didn't work well for "fine control". We ended up with motor = joystick³ (or some other odd, greater than 1 power; it might have been ⁵). Then you fix, via coefficients, the endpoints of the equation such that the lowest joystick value hits the full-reverse motor setting, and vice versa for forward. The exponent dampens the curve near zero, so there's more "joystick space" spent on the very slow speeds for the fine control, but the operator can still ask for max speed if they want it.

(With joysticks, you also have to deadzone the middle, as "operator has centered the joystick" will sometimes report joystick values of 1 or -1, instead of 0. Or buy a better joystick.)

I'm not 100% sure I'd apply this to a shower: max hot on my shower is hot, I don't usually ever want it. And I wouldn't want to scald someone.

For a long while, our shower was not the typical tiny goldilocks zone surrounded by ice and lava: … it never really got above "lukewarm", even at max. We had the shower valve replaced and it works better now. (And, apparently, as I learned, there's a mixing valve at the hot water heater: i.e., the hot to the shower is not direct from the water heater, like I had imagined. We raised the temp there, slightly.)