I came across a really interesting type of record called "Kniha sirotčích" or "Orphan Book." This kind of record may or may not exist for your locality. You will find it with other land records and town records - the "Town Books" - which are usually stored in one of the five Czech Republic city archives. These are located in Brno, Ostrava, Plzeň, Prague, and Ústí nad Labem.
According to the familysearch wiki page about Czech Town Records:
Here is a direct link to the "Kniha sirotčích pasiv pro Novou Bělou" from 1828-1850.
Sadly, the family that I am researching had already moved from Nová Bělá to Kunčice nad Ostravice, or what was then known as Gross Kunzendorf, by the time the illegitimate birth occurred. So, my record of interest was not found in this book. But here is an example of one from this book:
Transcription to come.