Here, I explain the problem of defeasible justification.* This is a philosophical puzzle that applies to almost all our knowledge & justified beliefs (all those that have fallible justification); Hume’s problem of induction and the problem of external-world skepticism are special cases. (Caution: what follows is pretty abstract-logicky.)
I think the second solution still leads to infinite regress. Take n to be the necessary truth that provides reasons for rejecting h'. Now, n must provide either defeasible or indefeasible reasons for that. If first, we need to provide a defeasible justification for (n entails ~h'). If second, both n and (n entails ~h') are necessary truths and thus ~h' is a necessary truth, but we assumed it is contingent.
I think the second solution still leads to infinite regress. Take n to be the necessary truth that provides reasons for rejecting h'. Now, n must provide either defeasible or indefeasible reasons for that. If first, we need to provide a defeasible justification for (n entails ~h'). If second, both n and (n entails ~h') are necessary truths and thus ~h' is a necessary truth, but we assumed it is contingent.