When should ¹/EFL become ¹/(2 × EFL) to avoid camera shake?
When you plan to display at twice the magnification needed to create an 8x10 print to be viewed at distance of 10" by a person with 20/20 vision.
The original 1/FL rule of thumb was based on the assumption of a 36x24mm frame of film being enlarged to an 8x10 print. Even then, if you were planning on printing twice as large many photographic textbooks recommended shooting at 1/(2xFL). The same was (and is) true of depth of field. If you are printing twice as large you need to halve the size of the circle of confusion, which results in DoF half as deep for the same focal length, aperture, and subject distance.
In the current environment, if you are displaying a 24MP image at 100% (one screen pixel per one image pixel) on a 96ppi monitor (a typical 23" 1920x1080P HD monitor, for instance), you're looking at the equivalent of a section of a 60x40 inch print! In that case you would need to use the 1/(6xEFL) formula to reduce the same amount of camera motion allowable for an 8x10 display size to be undetectable.