Have archeologists ever recovered any material from ancient times that promoted atheism or non-secular thoughts? If not, when was the first case of atheistic views found in history?

Not exactly archeological, but one of the important characteristics of the Milesian school of Phylosophy, around the 6th or 5th century BCE, was an approach at explaining the world through physical phenomena as opposed to invoking the action of the gods as explanation.

A Greek poet, Xenophanes of Colophon, very influenced by the milesians, was very critical of the naive, anthropomorphic pietist view of the gods held by the popular Greek religion, promoting a more sophisticated view.

One of Xenophanes' poems is often quoted as: "The Ethiopians make their gods black and snub-nosed; the Thracians say theirs have blue eyes and red hair." and that if oxen could draw their gods, they'd have tails, hooves and horns, implying that the gods as the Greeks conceived them were the self projection of the Greeks themselves into divinity.

Also Anaxagoras of Clazomenae is one of the first philosophers to design a "physical cosmology" describing the origin and constitution of the universe without recourse to divine intervention. So, he described the celestial bodies as physical objects, made of rock, molten metal and other physical materials.

However, it's not adequate to describe those philosophers as atheist in the modern sense of the world. Even Xenophanes and Anaxagoras participated in Greek religion and weren't out there to convince people not to sacrifice for Zeus or not to participate in public rituals.

It's very complicated to understand this with modern eyes, because religion means a very different thing for us than it did for the ancient Mediterraneans. Religion was not conceived as personal spirituality or a relationship with the divinity, but as a cultural, national, societal manifestation. It was intertwined into political activities, military activities, civic manifestations and even in the way people had fun. Even the theater festivals, when people went to the amphitheaters to watch a fun comedy by Aristophanes, was something of a religious activity.

So, there's that.