Nope...Moses started writing a couple of months after they left Egypt...
Exo_17:14 And the LORD said unto Moses, Write this for a memorial in a book, and rehearse it in the ears of Joshua: for I will utterly put out the remembrance of Amalek from under heaven.
And Moses continued to write during the 40 years as he was instructed...And Moses finished up here...
Deu 31:24 And it came to pass, when Moses had made an end of writing the words of this law in a book, until they were finished,
Very good!
Exodus 24 records Moses’ writing. He recorded the experience of the Israelites to that time.
One of the reasons the received text (as written in the official scrolls before his death) is not chronological and varies a little in style is because of the inclusion of earlier writing in the received text.
The Torah was actually ‘written’ in two parts; one both immediately before and after Sinai, and the second (the complete one we have now) at the end of Moses’ life. In that second part, Moses clearly commands remembrance of the lessons he himself had taught for the previous 40 years.
There’s actually quite a lot of traditional Jewish commentary on this topic.