Paul quotes most from Isaiah, The Psalms, Deuteronomy and Genesis.
Isaiah is an obvious choice with its large focus on eschatological hope, and also the inclusion of the gentiles.
The Psalms would be a good choice as the songbook of the Jews and the early Christians.
Deuteronomy’s emphasis on blessings and curses fits neatly into early Christian belief.
Genesis quotations are mostly to do with Abraham.
Unlike Philo who uses a metaphorical method in interpreting scripture, Paul does not have a simple system.
One interesting way Paul uses scripture is the way he changes the tense, e.g.
For the Lord will not forsake his people (Psalm 94:14)
God has not rejected his people whom he foreknew (Rom 11:2)
I shall not be put to shame (Isaiah 50:7)
For I am not ashamed of the gospel (Rom 1:16)
The way the tenses change implies that the decisive event has occurred.
Paul is not always bothered about the context of the original usage but is interested in the immediate application to this very important new situation.