That Mary is the new Eve is one of the earlier Marian doctrines that we see fully formed. It seems to me to be an important basis of many of the Marian dogmas. It was a belief that had spread out all over the known world by the mid second century.
I think that scripture attests to it, even if only in the more typological sense. The fathers saw many of the parallels between the story of Eden and the gospels.
Adam in the garden of Eden went to the tree to enact the fall of the human race, Jesus went from the garden Gethsemane to the cross, the “tree of life” to redeem mankind.
The fallen angel satan appeared to Eve, ultimately starting the chain of events that, because of Eve’s disobedience, would lead to the fall of humanity through Adam. The angel Gabriel appeared to Mary and started the the chain of events that, empowered by Mary’s obedience, would lead to the redemption of mankind through Jesus.
As St Irenaeus puts it:
And so the knot of Eye’s disobedience received its unloosing through the obedience of Mary; for what Eve, a virgin, bound by incredulity, that Mary, a virgin, unloosed by faith."
Furthermore, in Catholic belief, both Mary and Jesus were sinless virgins, as were Adam and Eve at the time of the test. It was sort of a divine do-over.
I would also argue that Jesus present in the Eucharist is the fruit of the tree of life. John 6’s “whoever eats of my flesh… shall have eternal life” seems to echo God’s concern in Genesis that man not eat from the tree of life and live forever. His body literally hung from the tree.
I’d also argue Peter drawing his sword in the Garden of Gethsemane makes him a sort of a figure of the angels who were ordered to guard the Garden of Eden.
A good resource is Cardinal Newman on “Mary as the New Eve”
christendom-awake.org/pages/marian/newman1.html
christendom-awake.org/pages/marian/newman2.html
The first part deals mainly with Mary as the new Eve, the second deals mostly with other Marian teachings.