Think about where you need each voltage. I'm betting either you don't need all of them everywhere to begin with, or components can be arranged with little effect on other constraints, to make exclusive regions where each is necessary.
Consider the size (relative number of pins) of each net. The more numerous ones are probably more generalized (evenly distributed over the board), too. These are good candidates for board-wide planes, or indeed the reference plane (GND itself).
It's generally a good idea anyway, to separate higher voltage / power nets, as transients/noise will be worse around them, and heat generation. These can be kept separate from low level signal/control sections.
Considering these points, you might not even need 6 layers anymore. Generally, 6 layers are only needed for very dense components like BGAs, high currents particularly with high complexity (more layers = more copper overall; with enough component density, fine pitch, or number of connections, that a heavier 2 or 4-layer build isn't practical), or occasionally for dense builds otherwise (perhaps with blind/buried vias or even HDI to allow components/pads to overlap).