It depends on how you define "reach". Technically speaking, when only factoring in the crop factor, the answer is "No, a cropped sensor like APS-C does not necessarily increase reach." Using a cropped sensor will definitely reduce your angle of view relative to a full-frame camera, as you are cropping the outer edges of the relative frame.
When it comes to discussing reach, what is more important is pixel density, or to be explicit pixel density in the context of a narrower angle of view. I'll get to the specifics in a moment. In most cases, APS-C sensors do have smaller pixels than full frame sensors. As such, historically, it has indeed been true that APS-C type cameras offer greater reach than FF, but with the advent of high density full frame sensors like the D800 and even more so Canon's 46.1mp prototypes that are supposedly out in the field, this is no longer guaranteed to be true.
An ideal comparison of FF to APS-C from a reach standpoint would be to compare the 18mp 1D X and the 18mp 7D. Both cameras have the same number of pixels, and the same image size output. The 7D truly has "greater reach" not only because it has a narrower angle of view, but specifically because it has much smaller pixels. The 1D X has 6.95 micron pixels, while the 7D has 4.3 micron pixels. For the exact same image area, in a photo taken with the exact same lens, the 7D will produce more detail. If the 7D also had 6.95 micron pixels, despite it's narrower angle of view it is not actually extending your reach. The 7D would produce 3208x2144 pixel images that contained the exact same detail as a 1D X image cropped to the same dimensions.
From a pixel-size standpoint, the 7D has about a 60% "reach" advantage over the 1D X, and a 45% reach advantage over the 5D III. If we compare the D800 with the 7D, the 7D's reach advantage shrinks to a mere 7%.