Yes, sure. I can get 32 pieces with 6 movements. And I do not believe 32 is the maximum possible number.

How to get 32 pieces? Think of the regular 6 cuts. The modification comes at the last one. Let it be the upper horizontal cut. Look at the bottom layer that has already been cut into 3x3 squares. Arrange these squares so that the 6^th cut - which, as a movement, is going to be the same as before - cuts the 3x3 squares as well. In a 3x3 grid of sqares you can draw a line cutting through 5 squares, which will give you 5 additional pieces.