Is it possible to double the resistance of a potentiometer?

I'm starting a circuit-bending project using the rhythm PCB from an old Viscount electronic organ. The idea is to make a portable rhythm box by adding some extra controls and processing. What this question is about is my plan to add a half-tempo switch, in addition to the existing exponential fader that controls tempo.

My initial thought is to add a switch that will double the resistance when activated.

I assume this isn't possible except by placing another variable resistor on a circuit path, as so:

^{simulate this circuit – Schematic created using CircuitLab}

This would require a design that forces keeping the two potentiometers physically aligned, though. If there's some trick to do this where one potentiometer will automatically 'track' the other one, I'd like to use that.

Edit: Per questions, here's a photo of the PCB:

And relevant part of the circuit diagram:

Answer

With the newly added original schematic, we can see that this is a RC oscillator circuit. To achieve your real goal of adding a half tempo switch, we can divide the tempo in half by adding a second capacitor with the same value as the original one. With the switch the resulting circuit segment would look like this:

^{simulate this circuit – Schematic created using CircuitLab}

When the switch is closed the capacitance is doubled, which means the frequency is halved. That sounds by far like the easiest option here - much easier than replacing the potentiometer with anything.