Dust your working space generously. Gently lift the first flattened dough ball. Let it hang for a few seconds so that it weighs itself down and stretches. Do this for all sides by moving the dough with your fingers like a steering wheel. Now transfer it to your working space and dab it with 1-2 tablespoons of oil (or butter). (If the dough has rested well, it will immediately stretch a lot.)
Take a hold of the dough on the left and right edges. Spread your arms gently, stretching the dough further in the process. Then do the same on the top and bottom sides. You'll now have a big circular dough shape that's thinner on the inside and a little thicker on the outside edges. You can lightly lift it, and wave the dough in a motion similar to draping the tablecloth over the table to stretch it further. Now continue to stretch it with your hands.
Move your hands (knuckle side up) under the dough gently, and continue stretching it by pulling it toward yourself. Pull in slow movements, and walk around the table (and the dough), pulling it on all sides a little bit at a time. Stretch the dough little by little, by walking in circles several times. It should be equally thin everywhere by the time you're done. The middle of the phyllo and the sides should be equally stretched.
Continue to stretch until the dough is of the thinness of cigarette paper. Additionally, if at some additional point(s) during stretching you feel the dough needs more oil, dab it in again, 2-3 tablespoons at a time. Finally, as you're spreading the dough, shape it into a large rectangle or a circle.
Try to avoid rips and holes as much as you can. If they do happen, work around them, or try to patch them by lightly pinching them together. (If the dough keeps ripping in a big way, read the article. You may have to start over!)
Note: if the dough has rested enough and you kneaded it well in the first steps, it's possible it will stretch even several inches beyond the table.