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Completed as in having added books in addition to Torah. How do we know that? Well, we have the five pre-Chrisitian era Law books and two of minor prophets. If they had minor prophets it makes sense to think they also had major prophets, and psalms which are very, very important part of Eastern Christian liturgy.

From Pauline Epistles, which were written circa AD 40-60 we have OT references which do not correspond to the Masoretic (aka "Hebrew" Bible or the targums), from more than just the Mosaic books and the two minor prohest, so there must have been material considered scriptural in Greek which included other parts of the Old Testament.

Other New Testament authors writing between AD 70-100 do the same thing. Philo of Alexandria, a Greek-speaking 1st century Jews speaks of the Septuagint, and so does the Jewish 1st century historian Josephus. Origen (latter half of the 2nd century) didn't doubt its existence, nor did any other Christian apologetic as far as I know.

So, there is ample evidence to conclude that some canon of Greek-language Jewish scriptures existed and were being used by Greekl-speaking Jewish diaspora. Exactly what that canon contained is not entirely certain, but obviously it contained some books not otherwise found in the Masoretic Text.