I was just wondering, instead of only having the maxSpeed at the exact midpoint, how could I get it to stay at the maxSpeed for the center 1/3 of the rotation?
Use a larger value for maxSpeed, then constrain the result of the map to be between lower (reasonable) limits.
What is it you are turning? What values have you picked for minSpeed and maxSpeed? How fast does that result in whatever you are actually turning moving?
With a valid range of only about half the potentiometer's actual range, it looks like you will only accomplish a 180 degree (approximately) rotation of whatever you are moving. I don't see how running full speed for more than a little of that range will make much difference in how long it takes to rotate that 180 degrees.