Many of us suffer from the same thing. I don't think we've found a definitive solution yet. Cider vinegar work for some, some have a few nuts before bed, Breakfast is supposed to help. One study on the phenomenon said that basal insulin might be the best solution at present, but my doc wasn't keen on that. I'm afraid you'll have to keep experimenting. You may find something that works.
As for the liver running out of glycogen (stored glucose) well that's the theory, but I've not seen any evidence of my liver doing that yet. I suspect I have a fatty liver though and who knows what is going on with the biochemistry in that. Remember the liver can metabolise glucose from protein, glycogen and fat.