Potentiometer partial range usage
Well, usually it's a very bad idea to use such a small part of a potentiometer element. There may be mechanical travel that has no electrical effect, and the position of the wiper will become extremely sensitive to motion, temperature, vibration etc.
Anyway, you can add an amplifier to the output. To take 61.8mV and output 5V, you'd need a gain of about +80.9. The op-amp should have low Vos for the results to be useful.
Even though this answers your question, I suspect this will be of no benefit to  you. You might want to explain your actual problem. If you are trying to get a measurement of a small displacement with a pot that has a large motion, this method will likely not be of much use unless  you've got something like a 10 turn helical coil pot with a hybrid element.