DIY water tank gauge using analogue ammeter - linearity
The usual way to correct the curves is to introduce a non-linear element, such as a diode. For example, diodes conduct practically zero current if the voltage is less than Vf (0.3V - 0.6V typical). So putting a diode and a resistor in parallel to one of the existing resistors would cause resistance to decrease once voltage reaches 0.6 volts.

Unfortunately, this method is not be applicable to your device, as your meter sees 0.06 volt drop, and no semiconductor has Vf that low.

If you really had to do this, I'd suggest using opamps to amplify the signals so they are in range for diode correctors, or just going straight to MCU. But this is probably going to be an overkill for your application, so instead, I think having a custom non-linear scale on the meter would be the easiest solution: