To assure an even background for the step wedge, it's easiest to have a sheet of translucent perspex, opal glass, or a diffusion theatre gel behind the wedge. Backing with a clear piece of material (or nothing) will always produce the problems you're encountering and create an uneven background for the step tablet unless you have a very evenly lit uniform surface behind it. Make sure no direct shadows fall on the back of the diffusion material you use, as that will cause local 'density' differences.
To avoid flare, mask off the area around the step wedge + diffusion material with darker material, perhaps a larger piece of foamcore, and use the lens hood you mentioned. You could also use a 5000K fluorescent lamp in a clamp lamp reflector behind such a set up to make something consistent and usable at any time of day or night.