Wall jump without returning to wall

In my 2D Platformer character controller that has both a wall jump and a double jump ability, how would I approach implementing a feature that stops the player from returning to the wall they initiated a wall jump from, while still making sure my controls feel responsive? (like they did in Celeste)

So far, I have got a timer that starts when the player initiates the wall jump and stops the player from moving until the timer is over. I’m already not too happy with that approach because it leads to this moment where it feels like player input is basically ignored.

Now in addition, I realised that after the timer is over, while just pressing left won’t be enough to return (to a higher part of) the wall, simply doing a second jump will enable the player to easily make their way up the wall.