The Saxton Hale code for jumping has been re-written multiple times. In its current form, it is possible to use momentum to power your jump farther and higher (if done right).
First off, lets give an easy example.
Someone's on the Hakurei Shrine Torii Gate, (more specifically a Demoman or Pyro, and sometimes Soldiers) and you just can't reach them by simply jumping as Saxton.
So what do you do? You use your momentum. The more movement you have the farther and/or higher your jump will be. A simple way to get more momentum is by climbing on the Shrine itself, then get near the edge that faces the Torii Gate. Once you are near the edge, give yourself a little takeoff space, hold right click, look up at about a 80 degree angle, almost straight up, then run forward, jump + crouch. At this point, you should be falling off the edge of the shrine, heading towards the ground (but you're looking up, remember). The following part requires the psychic ability of prediction and timing: You must let go of right click and crouch RIGHT BEFORE hitting the ground, if done correctly, your downward force will turn into upward force and you will jump and amazing length, around 1.5x more powerful and higher than the normal Saxton Jump. You will be able to land on the Torii gate and kick the campers' asses.