Many of the ancient texts used by New Testament scholars as sources for first-century Judaism as background material for the New Testament are actually either Christian compositions or were written long after the first century, or both, and insofar as reconstructions of early Judaism are based on them, those reconstructions are of dubious value.
In my new book The Provenance of the Pseudepigrapha: Jewish, Christian, or Other?, and elsewhere, I have argued that this should be our understanding of many Old Testament pseudepigrapha and perhaps even one or two Old Testament Apocrypha. I do not believe that a convincing case has been made for any book of Sibylline Oracles as Jewish, although there probably is Jewish material somewhere in them. Nor has a Jewish origin or early date been persuasively argued for the Testament of Job, Joseph and Aseneth, the Lives of the Prophets, the Testament of Abraham, the Story of Zosimus, Pseudo-Phocylides, the Life of Adam and Eve, or the Prayer of Manasseh. The Testaments of the Twelve Patriarchs draws on Jewish sources but is itself a Christian composition from well after the New Testament period. There are even some reasons to think that the Wisdom of Solomon in the Apocrypha could be a Christian composition.
As for rabbinic sources, even the earliest rabbinic collection, the Mishnah, was based on oral traditions and was only written down after 200 CE. The attribution of sayings to named (and therefore datable) sages is not reliable and cannot be used on its own to date a given saying. It is possible to show that some sayings are probably earlier than others on stratigraphic grounds and this in tandem with name attributions can allow us to isolate some early material. (In other words, if a group of sayings attributed to specific first-century sages can consistently be shown to be the basis of discussion for a group of sayings attributed to second century sages and not the other way around, there is a case for dating the first set of sayings in the first century.) In other cases, Mishnaic and Tannaitic ideas can be shown to be early because there are early parallels to them (but then the parallel already shows that the idea is early). But all this involves a lot of difficult spadework that isn't necessarily always done by people looking for New Testament background.
Related reflections here.