You do not mention whether there is a court order governing visitation. If there is no court order, each parent has the right to custody of the child. A court order would spell out when and if the child's father would visit him. It would also require him to make child support payments.
I'm not sure what you mean by "overnight" custody. If you mean immediate, the answer is no as there will have to be a hearing which takes time to schedule. If you are asking whether he can take baby home for an evening, that depends on the evidence presented in the hearing.
