If I understand you correctly, you are searching for a baby
in New Madrid County, Missouri. I would look for death certificate at Missouri State Archive Death Certificates under the Surname and the County. If that didn't work try Surname plus all the surrounding counties. Look for known parents and cemetery buried. Yes, I would open however many there are. The reason, you may find other children's deaths that you didn't know about. The great thing is when you click on a name and enlarge the death certificate and you go back the color of the box has changed, so it keeps track of where you are if you get a distraction.