  We present a variant of the solver in Zepeda-N\'u\~nez and Demanet (2014),
for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media.
By changing the domain decomposition from a layered to a grid-like partition,
this variant yields improved asymptotic online and offline runtimes and a lower
memory footprint. The solver has online parallel complexity that scales
\emph{sub linearly} as $\mathcal{O} \left( \frac{N}{P} \right)$, where $N$ is
the number of volume unknowns, and $P$ is the number of processors, provided
that $P = \mathcal{O}(N^{1/5})$. The variant in Zepeda-N\'u\~nez and Demanet
(2014) only afforded $P = \mathcal{O}(N^{1/8})$. Algorithmic scalability is a
prime requirement for wave simulation in regimes of interest for geophysical
imaging.
