Hello Throughfaith, the new birth is a part of the adoption process.
Ephesians 1
4 He chose us in Him before the foundation of the world, that we would be holy and blameless before Him. In love
5 He predestined us to adoption as sons through Jesus Christ to Himself, according to the kind intention of His will,
6 to the praise of the glory of His grace, which He freely bestowed on us in the Beloved.
Wait? What V23 is not the only place that our adoption as God's children is talked about in the Bible (obviously). In fact, let's take a look at another one of the passages that talks about our adoption (the preceding passage in Romans 8) to see what it has to say.
Romans 8
12 So then, brethren, we are under obligation, not to the flesh, to live according to the flesh—
13 for if you are living according to the flesh, you must die; but if by the Spirit you are putting to death the deeds of the body, you will live.
14 For all who are being led by the Spirit of God, these are ~sons of God~.
15 For you have not received a spirit of slavery leading to fear again, but you have received a spirit of adoption as sons by which we cry out, “Abba! Father!”
16 The Spirit Himself testifies with our spirit that ~we are~ children of God,
17 and if children, heirs also, heirs of God and fellow heirs with Christ, if indeed we suffer with Him so that we may also be glorified with Him.
Just like we are saved from the moment that we first believe .. e.g. John 5:24, we also become God's own sons and daughters by adoption at that very same moment. This is why, from that moment on, the Bible declares that "we are (present tense), the sons of God", that we are now able to "we cry out" (present tense) to God as our "Abba! Father!", and why the "Spirit Himself testifies (present tense) that we are (present tense) the children of God" (see again Romans 8:12-17 above).
This however, is simply the beginning of the story, not the end of it (for both our salvation and our adoption). For that we need to move ahead to the next passage in Romans 8, specifically to v23.
One of my commentaries on Romans has this to say (in part) about v23 and adoption (all of my commentaries come to the same basic conclusions, just FYI).
The “adoption” that we are said already to possess (Romans 8:14-17), is here made the object of our hope ( in v23). Christians, at the moment of justification, are adopted into God’s family; but this adoption is incomplete and partial until we are finally made like the Son of God himself (v. 29). This final element in our adoption is “the redemption of our bodies.” “Redemption” shares with “adoption” the (Pauline) “already-not yet” tension that pervades his theology, for the redemption can be pictured both as past and as future.
~Deut