As far as the Bible is concerned about the Nephilim, it seems quite clear that they are annihilated in the Flood, but somehow they survive. The Flood story starts in Genesis six and seems to indicate that the Nephilim were a major reason for sending it.
The Nephilim were on the earth in those days—and also afterward—when the sons of God went to the daughters of humans and had children by them. They were the heroes of old, men of renown. Genesis 6:4
As far as I know, the Bible never gives an explanation for how they survived. That they were giants is affirmed in Numbers and other places as well, the story of Goliath being most notable.
We saw the Nephilim there (the descendants of Anak come from the Nephilim). We seemed like grasshoppers in our own eyes, and we looked the same to them. Numbers 13:33
Whether these survivors are a separate race or the original Nephilim is not mentioned either. It is hinted, perhaps, that the Nephilim after the Flood were divided into plenty of tribes. Among these were Anakites (perhaps the first that brought about the others), Rephaites (Goliath was one of these and they were also called Emites), and Gittites. There's also the mention of some polydactyl Nephilim, having more than five digits on the hands and feet (1 Chronicles 20:6 and 2 Samuel 21:20). This is a major indication that there was something genetically different with the Nephilim.
The Nephilim are indeed mentioned and talked about much more in non-canonical books, but I'm not familiar with one that describes them as two races, one before and one after the Flood. If the early Church believed such a thing it was either speculative or based on other, non-Scriptural sources.
I've written an answer that covers pretty much everything the Bible has to say about the Nephilim, but it stops short of extra-Biblical sources, seeing that as off-topic for this site:
What were the Nephilim, and what role did they play in the Bible beyond just being mentioned?What were the Nephilim, and what role did they play in the Bible beyond just being mentioned?