  We investigate the asymptotic stability of solutions to the characteristic
initial value problem for the Einstein (massless) scalar field system with a
positive cosmological constant. We prescribe spherically symmetric initial data
on a future null cone with a wider range of decaying profiles than previously
considered. New estimates are then derived in order to prove that, for small
data, the system has a unique global classical solution. We also show that the
solution decays exponentially in (Bondi) time and that the radial decay is
essentially polynomial, although containing logarithmic factors in some special
cases. This improved asymptotic analysis allows us to show that, under
appropriate and natural decaying conditions on the initial data, the future
asymptotic solution is differentiable, up to and including spatial
null-infinity, and approaches the de Sitter solution, uniformly, in a
neighborhood of infinity. Moreover, we analyze the decay of derivatives of the
solution up to second order showing the (uniform) $C^2$-asymptotic stability of
the de Sitter attractor in this setting. This corresponds to a surprisingly
strong realization of the cosmic no-hair conjecture.
