"Fine tuning" a potentiometer
If I understand you correctly, you want to adjust things so the range only covers between ~30% and ~60% of the current range.

Let's say the output voltage from the pot wiper is currently 0-10V, here is a circuit to alter it to 3.3V to 6.6V (so 33% and 66% of original range):



The 10k&Omega; pot is represented by the two resistors in the dotted rectangle.

Output simulation with pot swept from 0 to 10k&Omega; (x-axis, output voltage on y-axis):



You can make it even easier to fine tune by using a 10-turn pot.