I haven't personally seen the new "red" cover which is delivered these days with new devices.
The old "brown" leather cover (from spring 2008) is very flexible at the hinge and puts no stress at all on the reader. It is designed to fold to the back. I almost always use it folded back.
Perhaps work the cover hinge a little bit to make it more flexible ?
But be careful if the material itself is too stiff it could break with too much usage (??).
If the material is genuine leather you could perhaps get some sort of leather "lubriant" to soften it up.