We know these books were in the Septuagint in Jesus’ day, since He and the New Testaments writers used terms to describe these Old Testament books, like “It is written,” “Have you not read?”, “the Scriptures say,” “the Law & the Prophets,” etc.
The expression “the Law and the Prophets,” which occurs several times in the Gospels, refers to the Jewish canon as it stood at that time, before the third part, “the Writings,” was added many years later.
That doesn’t mean, however, that individual books in that third section were held to be unsuitable in any way. The first book in the Writings section is Psalms, frequently quoted in the NT, even though, as far as anyone knows, it had not yet been formally incorporated into the Biblical canon in the time of Jesus.
The Law (
Torah ): Genesis, Exodus, Leviticus, Numbers, Deuteronomy.
The Prophets (
Nevi’im ): Joshua, Judges, 1 & 2 Samuel, 1 & 2 Kings, Isaiah, Jeremiah, Ezekiel, and the Twelve (the minor prophets from Hosea to Malachi)
The Writings (
Ketuvim ): Psalms, Proverbs, Job, Song of Songs, Ruth, Lamentations, Ecclesiastes, Esther, Daniel, Ezra, Nehemiah, and at the very end, 1 & 2 Chronicles.