I used something soft under most of my tanks. Just in case a small piece of gravel or dirt got under it, or the surface is not level. I do it for peace of mind.
I don't know how well the edges of regular construction styrofoam would hold up if only extending 1/8" beyond the frame. The rigid foam insulation or laminate flooring underlayment others have recommended might work better."Actually, done right, the styrofoam should be visible. As others have pointed out previously, it needs to be cut to fit under the entire tank, including the frame. You can paint the edge of the foam before positioning everything so it blends in with the frame or stand. "
The reason I said I would not see the Styrofoam is because the stand I was looking at has wood trim around the bottom of the tank that is high enough to cover the Styrofoam. This trim would also limit the size that the Styrofoam could be cut (on three sides) as the trim is only slightly set back from the tank itself (maybe 1/8").
The Styrofoam could fill out the stand area but would barely extend out past the edges of the tank itself. I am not sure if that matters or whether it would be better to have a stand without any border trim which would allow a larger piece.