Depending on the time and place, you might get lucky and find birth announcements in the newspaper. It has been my experience that the earlier in time I’ve been looking, the less likely I was to find those birth announcements. I suspect it was because of high infant mortality and perhaps not wanting to announce the birth if the child was not going to live. However, I have found some early birth announcements, long before vital registration was required by state laws.
This example is one that was published probably because of its newsworthiness, having had triplets! But the genealogical information in this little announcement is both sad and helpful at the same time.
Keep in mind, the children may not have been named at the time of the announcement, so you may see announcements such as “a son born to Mr. and Mrs. Aaron Businger.” But based on your other research, you should be able to identify that child based on dates or birth orders.
Have you looked for your own birth announcement? Here’s mine: