I would direct you to 'A History of the Japanese Language' (Frellesvig, 2010, p245) to see an explanation with examples. I must admit that his explanation is rather technical, but I have tried to paraphrase below.
According to the above source, the usage of as a conjunctional particle emerged in Early Middle Japanese (EMJ). As you mention, had previously been used as a subject marker. This subject marker function evolved to apply to cases in which a sentence had two clauses, one of which was a "headless nominalized clause" (a type of relative clause). Where would previously have marked a noun head as the subject, it began to mark these headless clauses. Frellesvig claims that we can interpret the usage in two ways: (a) as functioning to mark a type of relative clause, or (b) as joining two coordinate clauses. It is (b) which we recognize as the conjunctive marker function that we see in Modern Japanese.