Inostrancevia was a genus of gorgonopsid, living in what is now Siberia during the Permian period. This area was a hot, dry desert during this time. Close relatives of the ancestors of all mammals, gorgonopsids very likely had leathery, glandular skin, and also hair, though likely not in the role of thermoregulation. Gorgonopsids were fox- to lion-sized carnivores with well-developed nasal turbinates, indicating an active hunting lifestyle, though not necessarily a well-developed sense of smell. They did have a vomeronasal organ, though, and would have hunted by scent much like a monitor lizard. Inostrancevia was the biggest known, with a skull about two feet in length. Here, I've hypothesized a role for an early wet nose, coupled with ancestral hair follicles acting in a possible precursor role of guards/wicks for mucus and sebaceous glands. In my hypothetical restoration, mucus-wicking hairs poking out of the ancestral philtrum (cleft upper lip of modern mammals, less developed here and hypothetically comprising two parallel grooves) would collect airborne scents, which are collected by a flick of the tongue and passed across the vomeronasal organ to be detected. Glandular secretions of oily, waxy sebum (or a related substance) would have kept skin moist and pliable where flexibility was needed more than the toughness of cornified, callused skin. Fur was unlikely at this time, becoming more useful to later relatives in nocturnal habits as therapsids in general became marginalized by the upstart archosaurs (dinosaurs and kin).
I'm inspired by modern naked-skinned tropical mammals like hippos, rhinos, and Xoloitzcuintle dogs