Very few, if any, philosophical systems are really presented in an "axioms and their consequences" manner. I'm going to read the question as asking if their are any systems which provided arguments for the existence of a soul in a manner that isn't question begging. (Whether those arguments are satisfactory or persuasive is a different issue.)
A lot hangs on just what you mean by "a metaphysical soul." Depending on what you mean, these may not meet the bill.
Descartes argued that we could not rationally doubt the existence of our minds (which he identified with our soul), but we could doubt the existence of our bodies. On this ground, he took the mind to be metaphysically distinct from the body.
Plato, chiefly in the Republic and the Timaeus argued that we cannot know anything about the physical world as it is in flux, but can know only about the forms. In the Meno, he argued that "learning" is actually recollection of the forms, where we recall what we once knew before the trauma of birth so shocked our persistent soul that we forgot our former knowledge.
Neither of these are straightforward arguments from n premises to the conclusion that souls exist, but then in the context of a philosophical system, one rarely gets such extractable arguments for key claims. Both arguments for the existence of souls are, of course, the subject of hundreds of years of critical commentary. There are also many other examples that could be provided. (For instance, the Stoics held that souls were necessary for awareness.)