What's it called when bodies release oils after you wake up?
I am not sure what you mean by this, but our sebaceous glands are responsible for oil production in our bodies. They secrete throughout the day, not just at night. Sebaceous glands are usually attached to hair follicles and release a fatty substance, sebum, into the follicular duct and thence to the surface of the skin. The glands are distributed over the entire body with the exception of the palms of the hands and the soles of the feet; they are most abundant on the scalp and face. You can see an image of the glands on the skin below: