Character Controller Platformer - Hang on to ledges

I'm making a 2.5d style platformer where the big mechanic is that you can hang on to the edges of platforms by pushing against them.

For example, If you jump and go right onto the side of a wall, and hold D or right, you hang there until you let go or jump.

I'm using a character controller - but I can't find any way to do this. Here's my code right now:

 import UnityEngine
 
 class CC_Controller (MonoBehaviour): 
     
     public speed as single = 10.0
     public gravity as single = 8.0
     public jumpSpeed as single = 8.0
     private movey as single = 0
     
     moveDirection as Vector3 = Vector3.zero
     
     def OnCollisionEnter(collision as Collision) as void:
         for contact as ContactPoint in collision.contacts:
             localcontact = transform.InverseTransformPoint(contact.point)
             if localcontact.x > 0:
                 if Input.GetAxis('Horizontal') > 0:
                     movey = 0
     
     def Update() as void:
         controller as CharacterController = GetComponent[of CharacterController]()
         
         moveDirection = Vector3(Input.GetAxis('Horizontal'), movey, 0)
         moveDirection = transform.TransformDirection(moveDirection)
         moveDirection *= speed
         
         if controller.isGrounded == true:
             movey = 0
             if Input.GetKeyDown('space'):
                 movey = jumpSpeed
         
         if controller.collisionFlags == CollisionFlags.Above:
             if movey > 0:
                 movey = -movey
 
         movey -= (gravity * Time.deltaTime)
         moveDirection.y = movey
         controller.Move(moveDirection * Time.deltaTime)

This is my latest try using OnCollisionEnter, but that doesnt seem to do anything at all.

Is there a good way to do this?