The pot does not control the discharge of the timing capacitor. It is simply being used as a potential divider to provide an input signal to pin 9. Some calculations, based on the pair of 10K resistors in series with the pot, show the signal can go between 1.1V and 11V. (Here I am assuming the two 10K resistors either side of the pot in the images are R10 and R16 in the datasheet).
Adding a larger pot will not help as the higher resistance decreases the lowest possible voltage on pin 9. You won't get much lower than 1.1V!
You can, however, keep the pot and simply change the resistor value of R11 to a higher value. This will not affect the charge time of the cap C4, as the resistor is in parallel when the cap charges, but in series when the output of the third opamp disappears. (There is a diode, D2, that prevents the cap from discharging to ground via LED1)
Doubling the size of R11 will give you double the amount of time (give or take). I don't know the resistor value in that kit so you will need to figure out what you need!