He could do a "surrender" or a "waiver of interest" but you are getting a little ahead of yourself. He cannot sign rights over to a "boyfriend". It would have to be a step parent. So, if you are serious about your family, get married, then have this conversation with your child's father and your new husband. Paternity would not necessarily have to be established.
Please note that my answering this question, does not, in any way, mean I represent you for this, or any other case. You need to seek that actual face-to-face advice of any attorney in your area who can further explain the law as it applies to your case.