I use a pot now and from 0 to say 25% of the pot controls the full range of light. The rest (26%-100%) does nothing.
In this case, you'd measure the wiper resistance value at something like 30% of the pot and then use another potentiometer that's around 30% of the original value.
For example, let's say you start with a 4700 ohm potentiometer and 30% is 1410 ohm ... so you can use a 1500 ohm potentiometer or a 2000 ohm potentiometer.
As for controlling a LED, you don't control a led by adjusting the voltage. LEDs are current driven devices, you must limit the current going into the LEDs otherwise they burn out.
A LED with light up when the voltage goes above a threshold, which is called forward voltage. This threshold varies from led to led, but it's typically around some value, depending on chemistry used.. ex 1.7v for red, 2.2v for yellow or green .. 3v for blue
This threshold also changes slightly with temperature, a warm led will have a lower forward voltage than a cold LED.
Then, the LEDs have 3 regions ...
1. below the forward voltage where the LED basically doesn't produce light, doesn't turn on.
2. a few tens of a volt around the forward voltage threshold, the LED will start to produce some light, but it behaves like there's a resistor inside the led, it resists current flow
3. above the previous region, the led is fully open and will light up as bright as possible... and if you don't limit the current it will burn itself out.
So if you use the potentiometer as a voltage divider, to turn on the LED, you're doing it wrong. Depending on temperature and actual led, the actual potentiometer value will be different, and the led will be fully on only in the 3rd region.
If you use potentiometer to limit current ... well, a led will be bright even with 0.5 mA ... it's kinda hard to limit a device at such low thresholds.
So what can I say ... I guess it depends on the circuit you actually use... maybe show the circuit so we can tell you where you go wrong.