It sounds like you might be pressing your finger at the wrong angle. Try rolling the index finger towards the nut (so your pinky rises away from the fretboard). You should be able to find a flatter part of the finger.
Unfortunately, the sweet spot is likely to be even more tender than the "center" of the finger. So it'll hurt more at first.
Edit: I feel compelled to remark (confess?) that I received this advice myself a few years after I started playing and it has made a world of difference for me. A year later when my thumb muscles developed, I was finally able to play Radiohead's Creep without resorting to root-position, "open-voiced* chords (to rest my hand).
Even now, when I flex my thumb to see the muscle bulge out, I call it my Radiohead muscle.