The re-birth by the Holy Spirit is a second birth into a life Adam formerly had and lost in the fall; i.e., God's divine eternal life within Adam's immoral spirit, and now we are born without it, in spiritual death (which is not death of the immortal human spirit, but absence of divine eternal life within the immortal human spirit).
That divine eternal life of God is re-imparted to the human spirit in the re-birth (born again) by the Holy Spirit.
The human soul and human spirit do not occur separately in the NT, they are always together in the human person until death.
They share some of the same functions (emotion, rejoicing), and can only be distinguished there, not separated.
In reference to your Scripture above: in context of the resurrection and the resurrection body in 1 Co 15:35-58 and the contrast between the natural (sinful) and the spiritual (sinless) material body in the two Adam's, where
the first Adam had a natural body from the dust of the ground (Ge 2:7), and through whom a natural body is given to his descendants, and
the second (last) Adam, Christ, the life-giving spirit (Jn 5:26), who through his death and resurrection will at the second coming, give his redeemed people a spiritual body--physical, yet imperishable, without corruption, and adaptable to live with God forever (Php 3:21),
it does not relate to our new birth, but to our two different bodies, pre- and post-resurrection.