If the child's father is giving up his rights to the child, he would also need to sign and submit a form to the court. This form is often referred to as a "Consent to Adoption" or a "Surrender of Parental Rights" form. The specific requirements for this form also vary by state, so it is essential to check the local court rules and procedures.
The child's new wife does not have any legal standing in this process unless she has adopted the child. If that is the case she would have the rights and protections of a legal parent. If she has not adopted the child, she may attempt to seek visitation or other rights through the court, but this would be a separate process from the mother's petition for full custody.
It is important to note that custody disputes can be complex and challenging. It is advisable to seek legal advice from a qualified family law attorney to ensure that all necessary steps are taken to protect the child's best interests and comply with the relevant laws and procedures.