I added a precision hereafter, because my formulation
"S' doesn't stop at vt" could have been wrongly understood by SRists.

SR Story
________

The time LT t' = gamma(t - vx/c^2) can be written

t' = t/gamma + [tau/gamma - gamma tau], where

tau = (x-vt)/v

Let's remember that x is the position of some event
reltively to the origin of a frame S considered at rest
wrt a frame S' moving at v, and that t is the time at
which the event takes place, measured on a clock situated
on S.

According to SR (Cf. Daryl McCullough), a clock at
event (x,t) that is at rest in S', synchronized with
the rocket clock according to S' will show time
t' = gamma (t - vx/c^2).

This is an awkward interpretation of t'.
The reality is much simpler:

After a time t, S' is at a distance vt from S, so t' = t/gamma.
SR wants to take x into account, so it considers
that S' continues to move until x.
Then t' = t/gamma + ((x-vt)/v)/gamma.
But even SR knows that it had to deduct something from t',
if it wants to keep t, the only relevant time, into account.

As S' travelled the distance x-vt, the "something"
is logically ((x-vt)/v)/gamma, thus one is left with
t' = t/gamma. Of course, first adding then substracting
the same value would be silly, so SR had to find another
solution.

No problem for SR, as it considers that when a man
is killed by a bullet, the murderer is not necessarily
guilty, because the man could as well have killed himself
by running at, let's say 3000 km/h, against the resting
bullet.

Hence, the "something" is not ((x-vt)/v)/gamma (S' moves
wrt S, the murderer killed the man), but ((x-vt)/v)*gamma
(S moves wrt S', the victim killed himself).
Logic, which imposes internal coherence in
a formula (meaning here that either S' moves wrt S,
*or*, not *and*, S moves wrt S') is pre-SRian.

Brave "still" new SR world! 

Marcel Luttgens