In response to a question in a PM that asked whether patristic literature had more in common with the New Testament Epistles than with the Synoptic Gospels, I'd like to offer this quick charcoal sketch of a few of the main features of the later texts.
The Passion Narratives, the climax of the Gospels inspires / gives rise to the genre of Martyrdom literature. The basic elements are that the innocent saint is arrested tortured, makes confession that Jesus is the Christ and then dies some horrible death.
The Life of Jesus is in effect 'paraphrased' in the Hagiographical literature, the lives of the saints. There is a similar mixture of life and wisdom / sayings as there is in the stories about Jesus. The Sayings of the Desert Fathers are more like the Logia of the Gospel of Thomas.
The homiletic style Epistlolography of the New Testament is reflected in both letters written to churches and great tomes of sermons.
Duscussions about the Christian faith in the Epistles adopt a more logical and systematic approach. The brief credal verses give rise to much longer more ridgidly defined and carefully worded expressions of faith.
Ethical discussions about how Christians ought to live, and the measures taken to deal with offenders are formalised. Sexual morality and conduct continue to be discussed along with other social issues.
Expositions of the Old Testament in the light of Christ's revelation go viral, especially after Origen. Then are also commentaries on New Testament works.
'One of their own poets...' takes off as apologetic literature and at various times classical learning is used / interpreted in such a way as to fit the model that everything led / leads to Christ.
Apocalypticism also plays a small part in the later / post-New Testament literature.
I think that's a complete list of the genres that develop from the various pre-genres in the New Testament.