Did ancient philosophers have a method to align themselves with the Divine realms (or Spirit)?
From what we know about them, some of them certainly did. Part of this method consisted of studying the philosophical and poetic works of their forefathers. They assembled in Mystery Schools of old, where the spirit of the group took them to new levels of understanding by inspiration and, of course, exchanging their ideas about the Good, the Just and the Beautiful. Plato is well known for his use of his dialectical method.
This method (as used by Plato) presupposes certain basic insights into the nature of things, like the presence of subtle realms and orders of Gods, Heroes, and the like. Nowadays, it seems, large portions of inhabitants of the West seem to have lost these basic insights. But that is not the topic of this posting.
The Neo-Platonists combined Plato's teachings with elements of Aristotle's writings, like his work on logic. Proclus demonstrates the use of logic, and dialectical method, in combination with philosophical questions. Just study his tome "Commentary on the Parmenides" to find a rather lengthy example of this. He, like Plato, was very enamored by mathematics as a science and study of it as a first step to intellectual enlightenment.
So, philosophers of old had their ways and methods of reaching a deeper understanding of the world. Their knowledge and method was more of a philosophic-spiritual nature than directed to empiricism, but this is not to downgrade their insights. Their understanding of the essential unity of the world certainly seems larger to me than that of many of our contemporary philosophers.
Also, Pythagoras is said to cure depressions of his disciples through the use of music. He must have learned this either from someone else, or have discovered this method of curing himself, which likely involves empirical data, experience, and/or intuitive insights. His work on musical theory is well known, and involves mathematics as well as practical work on instruments.
Today, we have the advantage of the internet to exchange ideas. Imagine what possibilities this would have opened to them!