How do I alter the resistance floor of a potentiometer without shifting the entire range?
I essentially need to take a wild guess at how you're using your pot. The simplest passive(ish) circuit that I can think of, assuming that you are using the pot to sweep through 5V:







simulate this circuit &ndash; Schematic created using CircuitLab

You'll find that at the bottom of its range, the potentiometer will appear to produce a voltage flattening out to approximately that produced by 10deg deflection. There are much better ways to do this (i.e. op-amp circuit).

Another note: I assume that the next stage is high-impedance; otherwise, this is going to have problems.