Layers of collision, or multiple collisions

I am currently working on a platformer-type game. My game will need to recognize when the player runs into a wall (to initiate a wall-climb). My problem is that the player is usually colliding with the floor before they ever reach the wall. As such, the code used for colliding with the floor is being executed, while the wall collision is completely ignored. My assumption is because the OnCollisionEnter2D is being triggered by the collision with the floor, and can't be reset until the player "exits" the floor collision.

My question is, can I write my code to be able to have multiple layers of collisions running on one object at the same time (ie, the floor and the wall touching the player at the same time)? Should I place more than one box collider on my player, or is this impossible, and I should look at mixing triggers with collisions? Any feedback is much appreciated. Thanks.