  The universal traveling wave solution to the Balitsky-Kovchegov equation with
running coupling (and other equations in the same universality class) is
extended to subleading orders at large rapidity and small dipole size $r$. The
large rapidity expansion of the logarithm of the saturation scale $Q_s(Y)$ is
derived from that traveling wave solution. In addition to the two already known
leading terms in $Y^{1/2}$ and $Y^{1/6}$, which are determined only by the LL
BFKL kernel, the three following terms in $Y^0$, $Y^{-1/6}$ and $Y^{-1/3}$ are
obtained. They are universal and sensitive to NLL BFKL effects. Initial
condition dependence and NNLL BFKL effects would both start to appear only at
the next order, which is in $Y^{-1/2}$. In the light of these results, the
impact of the precise implementation of the running coupling in gluon
saturation is discussed, \emph{i.e.} parent dipole size prescription \emph{vs.}
Balitsky's prescription, and also the effect of the full NLL corrections, not
yet implemented in numerical simulations.
