Lowest travel distance for one finger? Of the layouts I know, that would have to be the standard cell phone layout. (edit: Correction, a telegraph keyer. Morse code wins for having NO travel distance and single finger input, just a lot more key presses) With one letter per key? Then you should look at layouts designed for... special needs. Maltron has a "Single finger or Head/Mouth stick keyboard".

But that might be taking things a bit too far. If you are looking to do something on a normal keyboard it would have to be one of the single hand Dvorak layouts.

Furthest travel distance? Two handed Maltron or Kinesis Advantage. Again, on a normal board... um, reverse one of the one handed dvoraks. Flip the top 2 and bottom two rows, then shuffle the middle 4 and outer 4 alphabetic columns, Finally shove the number and symbols columns down the center.

I usually use one hand when typing because I have  a cofee mug as a permanent extension of my body so I understand the need for keyboard layout that optimizes the movement capabilities of one's fingers

if I am doing one-finger typing, I'm not doing it long enough to care what the layout is.   If me some disability reduced me to it, I would hope my voice worked and would looking at using Dragon speech recognition in some form.