I have two ideas, out of which one is not very good.
Reinterpretation of direct-inverse morphemeEasily, the direct morpheme could be reinterpreted as solely being used when the subject is at the top of the animacy hierarchy, and thus either becomes a first person or first-and-second person congruence marker. (Some langs iirc rank second person higher, so that's also a possibility.)
This even leaves open the possibility of using the inverse marker solely as a marker for third persons, and then an unmarked verb could be second person. Other paths to such a situation can be constructed.
Unlikely rebracketing of case morphemesSome languages permit omitting the accusative marker on nouns when the subject is, say, a pronoun. (This might assume case marking on the pronouns still obtains or some type of congruence already in place - otoh, Chinese is somewhat pro-drop so why couldn't this work without a pre-existing congruence?)
Now, we can restrict this to, say, omitting case marking in the presence of a first (or second) person subject. See where I am going with this? Now, let's have the case marker - either a suffix or a prefix of the noun - condition a sound change at the word boundary of the verb or just be rebracketed as a subject marker, and then generalized to all persons.
SVO: an object with an object prefix triggers a change, causing a verbal suffix
SOV: an object with an object suffix triggers a change, causing a verbal prefix
Of course, a thing that could further influence this could be a split-ergative system, where absolutives and ergatives and accusatives cause different things to be rebracketed with different persons as subject.