not all tapered strings go down to the core, some go down to the last winding before the core. i use dean markley sr2000s, that are tapered on all strings bigger than the G string. also, tapered strings should be full size by the time the string gets 1/4" - 1/2" from the saddle, well before the pickups, or even the playable length of the string.
why taperwound strings? (repost from an earlier thread...)
taperwound strings intonate much more accurately than non-tapers, especially on the bigger strings. they provide a more concise witness point to the bridge, which in turn forces only harmonics that are closer to multiples of the fundamental to sound.
first off, some theory...
when an ideal, perfect string is plucked, only frequency multiples of the fundamental will sound. the fundamental is directly related to the length of the string, and all the harmonics are multiples of the fundamental, all in tune and sound pretty, regardless of how high they are.
with a real string, the witness point of the string to the bridge, along with the fact that real strings are not infinitely flexible like a "perfect" theoretical string, both work together to cause a sort of "gray area" for the string, where it is bending, not at a perfect point over the bridge, but more of a curve, due to the thickness of the string.
this gray area reflects in the fact that the actual length of the vibrating portion of the string is not clearly delineated. this causes higher harmonics to be present that are out of tune with the fundamental. a smaller witness point and a sharper break angle will cause this gray area to be smaller, and keep the higher harmonics more in tune with the fundamental.
a pic will help explain this better...
that's the main reason why tapers are better than non-tapers.