In De Trinitate Augustine interprets the handing over over the kingdom to the Father as not only the culmination of the mediatorial reign of the Son of Man but as resulting in the eternal blessedness of the saints: seeing God face to face.
For Augustine, the reality of the Son of Man in the judgement, a visible glory seen by the righteous and the wicked, it is surpassed by the glory that will exclusively be beheld by those who inherit the kingdom:
This contemplation is promised us as the end of all activities and the eternal perfection of all joys
It is of this contemplation that I understand the text, When he hands over the kingdom to God and the Father (1 Cor 15:24), that is, when the man Christ Jesus, mediator of God and men (1 Tim 2:5), now reigning for the just who live by faith (Heb 2:4), brings them to the contemplation of God and the Father.
Contemplation in fact is the reward of faith, a reward for which hearts are cleansed through faith...
For the fullness of our happiness, beyond which there is nothing else, is this: to enjoy God the three in whose image we were made.De Trinitate, Book 1:3:17-18