It would be in your best interest to be present at the June hearing. The court can order him to pay the past-due amounts, and can even incarcerate him and require a lump-sum payment in order to be released. But beyond this, the court cannot "force" him to pay anything. Courts grant orders of money owed, but have no part in actually collecting that money.
For better assistance with collecting the past-due child support, you would need to meet with an attorney. One could help you with drafting an income deduction order, locating attachable assets, and other mechanisms for collecting what you are owed. An alternative to using the assistance of an attorney would be to use the assistance of the Office of Child Support Enforcement. You would need to visit your local office to open a case, based on your existing child support order. In either event, you need to supply the individual (your attorney or your case manager) with a copy of ALL orders related to this debt (the original child support order and all outcomes of the abandonment proceedings).
Good luck providing for your child's needs.
~ Kem Eyo
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.