In order to address this question, I want to clarify that "supernatural" does not simply refer to ghosts and goblins and vampires, as we common think of it today. "Supernatural" actually has a much broader definition as anything that is outside of natural laws (so things that we cannot experience directly) and anything related to God or a god.
So, thought of in this way, there is much about this story that qualifies as supernatural. First, the veil itself is a literal symbol of a supernatural state; it is a tangible object that represents something intangible: the truth that humans are not only inherently sinful beings but that we all seek to hide the existence of our secret sins from one another. Second, the minister's belief that, after his death, it will no longer be necessary for him to wear the veil is a supernatural belief. We have no way to prove the existence of a life after death in the natural world; there is nothing that we can examine or point to as evidence of this eternal spiritual life. Thus, it too is supernatural. Third, Mr. Hooper's belief that God can see even our most secret sins, though we'd prefer to "[forget] that the Omniscient can detect them" is a supernatural belief as well because, again, there is no objective evidence in the natural world that can prove the existence of God.
In the end, the story doesn't offer us ghosts and werewolves but rather secret sins and spiritual life and God. Perhaps it is not quite as titillating, or -- for some -- perhaps it is more so.