I'm surprised no one has mentioned Alvin Plantinga, William Alston and Reformed Epistemology. Plantinga is recognized by even secular philosophers as one of the giants of the past century, and Reformed Epistemology has caused a major upheaval within epistemology.
The idea is that a belief in the existence of God can be "properly basic," meaning it can be rational and justified (or "warranted," in Plantinga's terminology) without any evidence or argument. Just as we implicitly believe in the existence of the past and the existence of other minds, we may simply believe in the existence of God.
In Plantinga’s version of Reformed Epistemology, a belief in the existence of God counts as actual knowledge because it’s produced by the operation of a special cognitive faculty designed to generate true beliefs about God. Plantinga calls this faulty the sensus divinitatis, borrowing the term from Calvin but not meaning exactly the same thing. Plantinga adds the wrinkle that one's cognitive faculties must be functioning properly.
Plantinga goes beyond mere theistic belief. Just as the sensus divinitatis generates a belief in the existence of God in a basic way, he says, the operation of the Holy Spirit makes the great truths of the Gospel directly known to the believer in a basic way. Voila, Christian belief may be rational and justified in a properly basic way, without any evidence or argument at all.
If this sounds silly and superficial the way I've described it - believe me, it isn't. Plantinga is a major, major force in philosophy, and Reformed Epistemology has generated massive amounts of discussion and debate over a period of many years.
The fact a belief in the existence of God may be warranted without evidence and argument doesn’t mean the belief is necessarily true, of course. The Reformed Epistemologists acknowledge this. Their point is simply that theistic belief may be rational and justified in a basic way, without all the evidence and argument we typically require when forming beliefs.
To Plantinga's notion of the sensus divinitatis, Alston adds the possibility that theistic or specifically Christian belief may be warranted on the basis of a profound mystical experience. Such an experience, of course, is a species of exceptionally powerful "evidence," at least for the experiencer.
Along the same lines, virtually all of the great sages and mystics of all religions and traditions have said that the divine may be approached only through intuition, which is regarded as an avenue to higher truths than can be reached through rational, dualistic thinking. Along the lines of what Plantinga says, they stress that what is gained through intuition is actual knowledge of the higher truths. Intuition operates most effectively through silent communion. In this circumstance, it may be difficult to distinguish between intuition and revelation.
A Christian classic along these lines is THE CLOUD OF UNKNOWING by an unknown 14th century mystic. His technique is to allow a "cloud of forgetting" to close the rational, dualistic mind and to enter silently and contemplatively into the "cloud of unknowing" where God can be approached.
I have said numerous times that I believe a well-founded belief system is the product of experience, observation, study, reflection and intuition. My approach has been to allow the "evidentiary" portion - experience, observation, study and reflection - to take me as far as it can, and then allow intuition through prayer and communion to confirm or disaffirm the positions I've reached through rational analysis. I thus do regard intuition as providing the higher knowledge.