Morality not based on a supernatural being does not necessarily leave only a morality based on the self. This is a falacy of a false dilemma. It ignores empathy, compassion, humility, and altruism which exists, arguably, in non-theistic individuals as much as theistic. Finding moral choices in how an act affects one's community, society, the environment, animals, offspring, children, family, etc. are also valid references for determining right from wrong that do not depend on a supernatural entity, nor the whims of the self.
I personally do not rely on a supernatural entity for determining right from wrong. This does not mean I'm making it up as I go according to my own whims, however. Objective methodology and criteria can be used for making such determinations, as is routinely done in free nations with the passing of laws, especially criminal laws, which statutorily define right from wrong. These laws, at least in the U.S., are not created and passed based on theological analysis, but on a far more secular basis - using data, evidence, history and reason - usually. In other words, it makes sense.
Shouldn't right and wrong make sense? Shouldn't these distinctions be mostly comprehendable regardless of where one chooses to believe the ultimate rules come from? For some of us, asking "why?" is absolutely necessary, and a "because XXX said so" is never going to be a good answer. This is because too many people around the world claim the same "XXX" for their authority of right and wrong, yet invariable cannot agree on what that right and wrong is. Thus, that eternal, unchanging standard is not only elusive, but exists only conceptually and without any tangible evidence such exists at all. At least it is not unreasonable to come to that conclusion.
This is not to knock those who choose to follow a particular supernatural entity and what they believe to be its determinations of right and wrong. It is just presenting another reasonable way of looking at the issue.