I keep hearing about the following as a way to keep as much as possible of the scene in focus: take the distance to the nearest object you want in focus and double it. This makes no sense to me. I understand focusing a third of the way in--that's somewhat supported by the dof tables. But how can this double the distance work when it doesn't take into account how far the farthest objects in the scene are? Hope someone can explain!

All that does is approximate that the nearest thing and the farthest thing are of equal focus, it does not guarantee that they are both appear to be in focus. That will depend on the aperture you select. If you take out a hyperfocal calculator, you will notice that the nearest point that the calculator says is in focus comes out to be half of the hyperfocal distance. The statement you made is simply the reciprocal of that. But, as you know, the hyperfocal point and the near and far point that will still appear sharp changes with aperture (assuming a fixed focal length of course)

As an example, if I want something that is 10 feet away and still sharp at infinity and I simply set focus to 20 feet and don't worry about aperture, it simply says that 5 feet and infinity will be of equal focus - that could be equally out of focus if the aperture selected for the focal length you are using does not support the needed DOF.