a good way to find presuppositions to base your logic off, is to attempt to understand reality from as many different systems of belief and presuppositions as possible first. then, eventually you'll discover conclusions that you cannot avoid no matter the presuppositions you choose. then, you can take those conclusions as your presuppositions to re-evaluate the evidence of your experience to construct beliefs.
From there, you can form a moral system that fits with your understanding of reality. Personal morals can never be "true", they are merely a reflection of our perspective. outside morals can never be better than our own, as morality is a concept exclusively within subjective existence. Theism is one possible presupposition, it's not more or less valid as a substrate for morality than any other- value judgments are inherently subjective as well.
If anything is, then it is itself. our representational and relational understanding of reality is always false, illusory. However, it's no less valuable than any other domain of reality- subjectivity is not good or bad, it similarly just is. We have to decide how to act, including the choice of how consciously we want to make that decision.