Stair climbing support for Third Person Controller (standard assets 5)

I am using the ThirdPersonController which is included in the Unity 5 Standard Assets. The character is based on a Rigid Body and a Capsule Collider and for the most part works rather well for my application.

However, it does not seem to support stairs properly. For very small ones it works to an extent (though in a rather weird jumping manner, and it gets stuck often) and for somewhat standard stairs it does not work at all (normally you can't climb them, and you have to forward-jump. The controller does not seem to support any air movement either, so you cannot even still-jump into the stairs).

What would be the best way to add or improve the stair-climbing support of the Third Person Controller? Should I just downgrade to the 4.6 standard assets, or create my own non-mecanim controller?