When you say .0013 taper, that's what is measured after taking a cut? Or that you've indicated using some kind of test bar? Do it taking a cut as using a test bar introduces lots of sources of error around how its mounted.
If you are not getting a parallel turn without tailstock support, there are three possible causes (at least that I can think of)
1) bed wear - causes the tool height to vary meaning its a different distance to the centre of the work, ergo different diameters.
2) the headstock is misaligned. Unlikely on a quality lathe unless there has been an epic crash,
3) the lathe isn't level. That means its own weight twists the bed such that tool is not the same height over the length of the cut.
99% its some combo of 1 and 3, likely (hopefully) mostly 3. The solution for 1) is recondition the lathe. No small feat, but a lathe worthy of it. The solution for 3), imo likely the bulk of the problem, is to level the lathe and then tweak with test cuts. Levelling is just a convenient way to compare each end of the ways for twist. You need a ,0005"/foot level, but if you don't have one, used what you have then do test cuts and tweak.
Don't level over the tops of the inverted V's, use parallels (proven) on the horizontal surfaces and get it same at the HS and TS (do your best to use unworn areas of the ways).
Test cuts. Put something substantial in the chuck, say 1 - 1 1/2" inch bar and turn it, light, fine feed etc. Measure the ends, and then tweak on of the the tailstock end floor bolts to remove the taper. For example, if it gets bigger toward the HS, raise the far side tailstock end mount 1/4 turn and repeat (or lower the near side). This can be an a periodic thing - concrete moves over time, and if bed wear is a big factor you sometimes have to tweak this to minimize taper on different work piece lengths.
My question is: is it possible that the headstock is not perpendicular to the cross-feed.
The HS is supposed to perfectly aligned with the ways, and the cross slide is supposed to be 1/2 a thou off of perpendicular to the ways. This is so facing can't produce a convex surface, just noting it as it might mess someone up who starting trying to measure whether the HS is perpendicular to the cross slide.
