There is a huge western bias to this question. For one, in English, our notion of and word for logic comes directly from Greek.

More importantly, the question implies that other global civilizations lacked logic, which is empirically untrue, or had a 'poorer' esteem for logic. It suggests that Greek science/math etc. was somehow more advanced than other groups, which is likely also highly debatable and similarly biased towards the Western tradition. One could easily argue that the Chinese had an equally strong appreciation for logic as seen in their respect for the Confucian classics. Ancient Mesoamericans were experts at mathematics and astronomy, but were the Greeks more logical?

Finally, the written record of ancient Greek thought comes from a very narrow range of city states. The Greek world had over 1500 city-states. There is no reason to assume that the records we have that are heavily biased towards the intellectual life of Athens are representative of the entire Greek world. Even just looking at the states we know, philosophical out look of Sparta was vastly different than that of Athens or Corinth. Were they less logical?

Do you mean logic as a branch of philosophy

Is this more of a question for r/philosophy? I don't know. /r/historianLA gave a decent response but I don't think it gets at the core of what OP was asking. It sounds like OP wants to get at the heart of the word logic rather than the strict definition and root of the word. I am also curious about the origins of rigorous reasoning or scholasticism of knowledge. What was necessary for someone like Socrates, Plato, or Aristotle to come around and start asserting things? What was deficient about the prior world that made reasoning and logic something that these people wanted to take and refine in the new world they were creating?