With your enlarger head all the way up, and a suitable lens, there should not be so much fall off... Are you adjusting everything properly? Make sure your light source is even (condensers adjusted or diffused correctly). You should try to project a much larger circle than the size of the contact print you are trying to make and then use a negative carrier or mask to get it down to size. That way, you'll only be using the center of the projected image circle and light fall-off should be minimal.
If you still have significant fall-off, then there are a couple of approaches. If you're making a negative and then contact printing that again with the same set-up, then the fall-off will cancel itself out for the most part, as Rob mentions above. You'll have to try and see if the results are acceptable.
I shoot a lot of black-and-white negative film with wide-angle lenses and no center filter. I compensate for this when printing by doing what I call a "center burn." I have a card with a lens-size hole in it. I start the exposure with the card almost resting on the paper and with the hole centered over wherever the optical center of the image is (with view-camera movements it isn't always the center). I then lift the card slowly toward the enlarger lens over a few seconds till the hole is close to the lens and the entire sheet of paper is being illuminated. I then whisk the card out of the way. For various lenses with various amounts of fall-off, I come up with a percentage of the total base exposure time for the "center burn." I've also learned that the movement of the card has to start fast then slow down. I've had good results with this technique. I do this at the beginning of the exposure, but it would work just fine at the end as well.
If you have a filter drawer above the negative stage on your enlarger, then you could experiment with diffusing materials either stacked in concentric circles or drawn on with soft pencil, etc. effectively making your own center filter for the light source. Many use a similar technique above the negative to "dodge" shadows (masking).
FWIW, edge burning does the opposite of what you want to do. You need an "edge dodge," which is essentially what my "center burn" does.
Best,
Doremus