  The organic charge-transfer salt EtMe$_3$P[Pd(dmit)$_2$]$_2$ is a
quasi-two-dimensional Mott insulator with localized spins $S$ = 1/2 residing on
a distorted triangular lattice. Here we report measurements of the uniaxial
thermal expansion coefficients $\alpha_i$ along the in-plane $i$ = $a$- and
$c$-axis as well as along the out-of-plane $b$-axis for temperatures 1.4\,K
$\leq $T$ \leq$ 200\,K. Particular attention is paid to the lattice effects
around the phase transition at $T_{VBS}$ = 25\,K into a low-temperature
valence-bond-solid phase and the paramagnetic regime above where effects of
short-range antiferromagnetic correlations can be expected. The salient results
of our study include (i) the observation of strongly anisotropic lattice
distortions accompanying the formation of the valence-bond-solid, and (ii) a
distinct maximum in the thermal expansion coefficients in the paramagnetic
regime around 40\,K. Our results demonstrate that upon cooling through
$T_{VBS}$ the in-plane $c$-axis, along which the valence bonds form, contracts
while the second in-plane $a$-axis elongates by the same relative amount.
Surprisingly, the dominant effect is observed for the out-of-plane $b$-axis
which shrinks significantly upon cooling through $T_{VBS}$. The pronounced
anomaly in $\alpha_i$ around 40\,K is attributed to short-range magnetic
correlations. It is argued that the position of this maximum, relative to that
in the magnetic susceptibility around 70\,K, speaks in favor of a more
anisotropic triangular-lattice scenario for this compound than previously
thought.
