DC Motor Speed Control Board
Figure 1. The area of interest in the speed controller.

Figure 1 shows that the speed control is a 5 kΩ (5000 Ω, not 5000 kΩ) and that it's output will increase from 0 to 12 V as it is adjusted from L (low?) to H (high?) speed. 


  ... if I move the pot about a quarter of the way around; the motor is already at the maximum speed I want.


It sound as though your drive is at full speed at 3 V out (25%). 
To increase the useful span of the pot we need to have 3 V out at full rotation.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 2. Modified speed controller. Options A and B.

It may be as simple as adding 15 kΩ in series with the top of the pot as shown in Figure 2, Option A. This will result in 3 V at the top of the pot. 

Alternatively put in a fixed 10 or 12k resistor and a 5 or 10k preset to allow you to tweak the max more closely to suit as shown in Option B.

You might see some non-linearity as the pot is loaded by R21, etc. If that's a problem we may need to scale all the resistance values down.



Just a caution: it might be worth checking what would happen should the pot go open circuit. Here's how to do it safely:


Disconnect the motor.
Measure the voltage on the W wire with the pot at mid-point.
Unsolder the wire from the pot wiper and measure the voltage on the wire (not on the pot).
If the voltage falls to zero then the motor should stop if the wire falls off.
If the voltage rises past 3 V (your maximum) then you need to consider if you need to put a pull-down resistor on W. You could try 100k or more and see if that's enough to bring it back down to a safe value.
If that doesn't work you may need to monitor the motor itself and put in independent speed monitoring and safety cut-out.