I ended up taking the right-side windshield garnish off and was able to peel away enough of the headliner to get my finger up behind the headliner to feel the holes.  The darning needle did help to locate the holes while I slid the bracket over the needle to maintain the location of two screws until I could get each screw started.  The third screw went right in place.  Since the garnish holds the headliner at this spot I shouldn't have an issue with the headliner once I reinstall the garnish.  I figure I can re-glue the headliner here if the garnish does not hold it properly.

Here's a couple of pics showing the right-side mounting with the garnish removed.  If the left side is different I'll post a couple of pics to show that location but it should be a mirror image of the right side.   I appreciate everyone's help on this.  The last thing I wanted was to put a hole in my new headliner that the bracket is not going to cover up!

I had the same problem ~ what I did was hold it in it's approximate position and I used a darning
needle to locate one of the screw holes. Once you find the Ist one, the other 2 are no problem.
Hope this is of some help...

That would have saved me some grief.  I'll definitely remember that for the next time.  Thanks!

Not that it helps now, but a trick I learned a long time ago is to put the screws in the holes before you put the headliner in, then you can feel where they are and cut a small X with a razor blade and push the material around the screw. Somehow feeling for those tiny holes is just darn near impossible. Sorry I can’t provide a picture, but try that next time, it’s pretty slick.

I recently replaced my headliner and am having a hard time finding the three screw holes to mount my sunvisor mounting brackets.  I failed to take a good picture of this before I put the new headliner in.  I've tried using a straight pin and my finger to locate the three screw holes and cannot find them.  If anybody has a 4 door 56' Customline could you please send me a picture of where your sunvisor mounting bracket is located on your car.  I'm thinking both sides should be the same.  Thanks!