The Hiddenness of God can refer to two different problems which are sometimes confused. The first is the question from a believer's perspective about the apparent lack of God's presence during suffering. Many of the Psalms address this from the first person perspective. The second has been formulated into an argument by J.L. Schellenberg. In this he argues that if God exists then God would make himself known; God has not made himself known, therefore God does not exist. It is this form of the problem that I will consider here.
One way that some Christian philosophers have responded is by denying that God would make himself known if God exists. The reason given for this is that it would violate human free will. The thinking is that if it is clear that God exists then humans couldn't help but believe in God and are therefore not free. This solution then connects up with the free will defense and says that God has permitted evil so that humans can freely chose him, therefore he purposely makes himself not clear so that this does not interrupt freedom. Interestingly, this view of God often appears in popular culture representations of a finite deity in movies and books.
In a previous post I've already discussed the problems with this definition of freedom (uncaused events) and this solution to the problem of evil (freedom is consistent with predetermination). Because of these considerations, it does not follow that the clarity of God's existence does interrupt freedom, or that if God's existence is clear then humans would believe in God. The alternative is that if it is clear that God exists humans can still fail to know God by failing to use reason to see what is clear. If humans do not seek then they will not understand.
Furthermore, the first premise of J.L. Schellenberg's argument seems to be correct. If God exists then this would be clear both necessarily and intentionally. That is, the works of God would necessarily reveal God (although humans could fail to see this), and God intentionally reveals himself through these works. This means that while I agree with Schellenberg's first premise, I disagree that it is not clear that God exists (his second premise). The philosophers I've mentioned above agree with his second premise but attempt to disagree with the first premise.
There is good reason to believe that if God exists then this is clear. Specifically, if humans are held maximally culpable for unbelief then there must be maximal clarity that we should believe. The need for atonement presupposes this kind of maximal culpability. For this reason, those who deny clarity also end up denying culpability and reinterpreting Christ's death as simply a sign of love, or a virtuous act, but not satisfying atonement.
I've discussed this in more detail in my book "The Clarity of God's Existence."