Just checking. I'm pretty sure this is impossible because of the icosahedron itself, but is it possible to fudge an FTI somehow so that, say, each face has nine or twelve possible rotational states and no moves are blocked?
Your suggestion would fundamentally change the geometry of the puzzle. It'd look like an Icosahedron and it'd have 20 grips but adding twist states to the faces would make it no longer icosahedral.
That said, the jumbling is a result of the angle that the axes meet each other so I don't think adding additional rotation stops to the faces will affect whether it jumbles or not. I also wouldn't call adding these rotations "fudging".