When I acquired the climbing axe, I tried climbing my first wall. Easy enough, just do as the game says: approach a wall, jump, hit the use/action key (E on the PC version) to attach to it, then move with the directional keys/buttons.
As an experiment, I tried jumping mid-climb, while still holding up/forward to keep myself against the wall, then hitting use/action again to reattach to the wall. Jumping seems to cover a significantly greater distance in the same amount of time than climbing normally, so whenever I need to climb straight up I just spam jump and E over and over again. Even if I need to move in another direction I imagine I can just do the same, only in that direction instead of up.
Now, there's a skill called Climber's Agility which I've gone ahead and mastered, which says:
Increase your climbing skills to traverse the environment with more speed and resist falling damage.
This would imply that the skill increases my climbing speed, but does anybody know if it's faster than the spamming-jump-and-reattach technique, if it makes no difference (i.e. spamming is still faster), or if it even speeds that up?