Conforming mesh with a flat plane

This is probably going to be a) painfully stupid/simple b) not even worth your time.
I’ve managed to build a small collection of additional mesh mods, but am having an issue with one aspect of conforming mods. My objective is basically “stickers” of various dimensions, but they don’t seem to want to be on the surface. Sub-dividing the edge/face is fine, because I can see that it curves with the body, but the texture is patchy/broken.
The crux of my question is: Using Blender, should I be adding depth to the plane making it a cube, or is using a negative offset on the Y-axis better, and, if so, what value (and is it better in edit or object mode)?

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