Finding my great-grandfather's illegitimate daughter
I'll start off by saying I know the answer may ultimately be "this is impossible." But I'd like to exhaust every avenue first.
According to what my mother and grandmother both told me, Grandma's dad had a child with a mistress and this child was a girl who drowned at the age of 3. I really want to know who my grandmother's lost half-sister was, and I feel like surely there's got to be a death record or birth certificate connecting them.
What I know:
--His name was John Clinton or Clayton Estep (b. 1899, d. 1944)
--He married my great-grandmother Mabel Todd in 1924
--He and Mabel had 7 children between 1925-1941, three of whom died in infancy (I have all 3 death certificates); their last child together was born August 12, 1941
--My grandma married August 31, 1942, and had a child in July 1943. I think my great-grandfather's illegitimate daughter had already died by then based on a story she told me that he warned her about leaving her newborn unattended (and she thought he was a blazing hypocrite to say so, heh)
--He died August 3, 1944, in jail
So the broad dates are this child was born and died between 1924 and 1944, but more narrowly I think she must have been born sometime between 1933-1941. I thought death certificates might be my best bet. No one has any idea what the mother's name could have been. Any advice on how to search for a child with an unknown name, unknown mother, who was no older than four at the time of death, and probably died before 1943? (Oh, and almost certainly this all happened in Ohio. Possibly Kentucky, but doubtful.) Thanks for your advice, even if the conclusion is "no way to search" :)