Placement of linear voltage regulators (L7806CV & L78M05ABV) and N-Channel MOSFET (P30N06LE) circuit for maximum power efficiency
Placing the FET before the 6V regulator definitely makes higher efficiency, because the regulator also consumes current when it's powered. Whether shutting it down makes a huge difference or not, it obviously depends on how much your 6V load is, and what is the quiescent current of the regulator, and what is the consumption of the rest of the circuit. If your 5V circuit consumes 1A, 5mA for the 7806 does not make much of a difference. But it's higher anyway, if you shut the regulator down too.

You can't power the 5V regulator from the 6V regulator, as 6V is not high enough voltage for the 7805 to work.