I think it would be cool if the base of the chess piece was a triangle instead of a circle and orientation mattered. So you could introduce a new move into the game which is picking up a piece and twisting it 60 degrees to put you on a different axis of the hexagon. This opens up three new directions (sides 2, 4, and 6) and closes the previous three (sides 1, 2, and 3).

Edit: Fix math.