John’s profile is rather limited. He also has a profile on FamilySearch/Family Tree which is pretty sparse as well.
Did John own any property? If so, a search of wills may show if it was inherited, and thus his father’s name.
Was John prominent in any way? Did he have a coat of arms? If so, check the Visitation(s) of Hampshire?
Did John leave a will? Was there an Inquisition Post Morten?
The same questions may be asked about his father, Oliver.