I have this same problem. Oily hair can leave rashes on my face and neck and even my chest and back.

Safe Soaps- Dr, Bronner's is very good. Bar soap that has been home made usually found at local farmers markets and contains all natural ingredients but make sure you read the ingredients. I use the coconut based soaps.

I finally just shaved all my hair off. Not what you want to hear, but consider the sacrifice of hiking the AT.