A child whose mother died at birth would have no bearing on whether or not the child belonged to the family. The father would have been under extreme pressure to marry again and given the times could easily have just left the child with a neighbor or friend and walked off (informal adoption). If the adoption were formal then you can look up the record in the county records where the mother died. It depends on the state as to which court handled this, its usually the District Court of Clerk of Court (just google your county) looking for family law, probate, guardianship. Then go to familysearch.org and look at their catalog under court records to see if they have the set of records that include adoptions and you can rent the film.

If the child was due to inherit then there may be a question about how this would work on a state-by-state basis because each state has its own laws on inheritance.

Interestingly, if the father had died back then and the mother were still alive, the child would still be considered an orphan and be provided by the court with a guardian. Go figure.