In my state, birth certificates have to be filed within 2 weeks, and the hospital where the woman delivers is responsible for getting all the information and filing it with the county. I don’t know how much time they have to file a death certificate. I would probably call the county and ask for the office that handles those certificates and they should be able to tell you about timing. It would be under her last name unless she might have given the baby a different last name (yours)?
Today, you could also consider calling larger more “regional” hospitals in the general area where she lives. If she arrived at a smaller hospital that didn’t have a neonatal intensive care unit, they would have transported her out to the closest center that did, if they thought she was stable enough to move.
One more possibility – some women will request that their admission be a “do not announce”. That means if anyone calls looking for information on them, the caller is told there’s no one there by that name.