Thanks for the feedbacks! I've made some progress on the animation, not yet finished but pretty close.

I included a subtle anticipatory motion, I can't really overdo it, it has to happen fast in-game, and be fluid. The hair are my favorite part, I exaggerated the motion to give the character more energy. Animation is a bit longer than the previous wip version, but still compatible with what actually happens with the game object. For those interested, I separated the hitbox of the object when climbing and the animated sprite, which is displayed at fixed coordinates during the climbing. It's clean, always in the grid, and I just have to make sure that the hitbox is where I display the sprite.
Any feedback welcome!
- Skaz
[EDIT]

Just finished this animation, this is visual feedback for the players, letting them know they can also jump directly on the opposite direction when holding on to a ledge. It's not a mandatory step to perform the jump, as I said, only visual indication. The end is a bit hard with the braid, I might rework it a bit, but I don't want to add several more frames.