You need three new animations for our character: EdgeGrab, EdgeGrabLoop, and EdgeGrabClimb. The first one is a transition from the InAir animation to the EdgeGrabLoop animation. It should have minimum movement in the root node if possible. The second one is a looped animation of hanging on the cliff's edge. The last one is an animation that uses root motion to climb the cliff's edge and ends with an Idle pose. See the following screenshot for reference:

EdgeGrab and EdgeGrabLoop animations should have all Bake Into Pose options selected in Import Settings. EdgeGrabLoop should also have the Loop Time option selected. You can also use the provided example Unity project and go to the Chapter 04 Character movementRecipe 08 Using triggers to grab an edge while jumping directory. You will find an Example.unity scene there. Open it, play the game, and press the space bar to jump towards the edge. After the character grabs it, press the up arrow to climb over the edge.