How do you display an image on top of another object?

I want to display an image on the ground underneath a character in the scene. The ground is flat. When I put the image in a quad at ground level with transparent shader, only part of it shows up, presumably because the renderer think that the ground is "in front" in places.

Putting the image slightly above the ground may not work because when you're far away, with reduced accuracy on the distance, the engine might see them as being at the same height anyway.

How can I display an image on the ground, that can move with the character?