1. meditation is not "unfocused thought" or "deep thought" intentional or not.
Meditation is "quit thinking" it is when you say NO to your mind and surrender yourself to pure feelings inside you. it can be scary
2. " I get a nagging feeling that there is something significant that I am unable to perceive"
you do not need to be a theist or religious to be a seeker. in fact my personal opinion about all religions is that they all are techniques to know "God" and by God i mean yourself as an undetachable part of everything.
3. and this is a real theory of mine (i have never tried it yet) i think dmt opens a door in to deepest places of your subconscience. and i will not talk about what i think those entities are which as Nexusians say you can meet in your trips
so i suggest Just keep searching and trying different techniques and methods of knowing yourself. do not say now to stupid things my friend