I saw another schematic with a resistor in parallel with the LED after the pot. I did not try that yet.
THAT circuit makes NO SENSE AT ALL.
DO NOT use that circuit for ANYTHING.
Unless you want to blow up the LED and/or the pot.
You still have not shown us your circuit with "a range from 2.9v to 3.10v"?
Why not. If the fixed resistor limits the current to the maximum amount the led can handle, the pot can dim the led. You would want a value that would give you the most usable range before the led drops out
You must not be looking at the circuit @vidarr was talking about. Let's review the schematic:
First, the resistor in parallel with the LED (R2) is completely superfluous and serves no useful purpose.
Second, the potentiometer will allow more and more (uncontrolled) current to flow as you approach zero ohms to the point where the LED will burn itself out unless the current is limited somewhere beyond the schematic we are seeing.
Third, the potentiometer may itself burn out as more and more current is shoved through it as it approaches zero ohms. Depending on the potentiometer, of course.