  We study the interstellar medium (ISM) properties as a function of the
molecular gas size of 82 infrared-selected galaxies on and above the main
sequence at $z \sim 1.3$. Molecular gas sizes are measured on ALMA images that
combine CO(2-1), CO(5-4) and underlying continuum observations. We include
CO(4-3),CO(7-6)+[CI]($^3 P_2-^3P_1$), [CI]($^3 P_1-^3P_0$) observations for a
subset of the sample. The $\geqslant 46 \%$ of our galaxies have a compact
molecular gas reservoir as these lie $\geqslant 1 \sigma$ below the optical
mass-size relation of disks. Compact galaxies on and above the main sequence
have higher CO excitation and star formation efficiency than galaxies with
extended molecular gas reservoirs, as traced by CO(5-4)/CO(2-1) and
CO(2-1)/$L_{\rm IR, SF}$ ratios. Average CO+[CI] spectral line energy
distributions indicate higher excitation in compacts relative to extended
sources. Using multiple molecular gas mass tracers, and conversion factors
tailored to their ISM conditions, we measure lower gas fractions in compact
main-sequence galaxies compared to extended sources. These results are
consistent with a picture in which mergers have driven the gas in the nuclear
regions, enhancing the CO excitation and star formation efficiency. We suggest
that the sub-millimetre compactness, defined as the ratio between the molecular
gas and stellar size, is an unavoidable information to be used with the main
sequence offset to describe the ISM properties of galaxies, at least above
$M_{\star} \geqslant 10^{10.6}$ M$_{\odot}$, where our observations fully probe
the main sequence scatter. Compact main-sequence galaxies are consistent with
being an early post-starburst population following a merger-driven starburst
episode, stressing the important role of mergers in the evolution of massive
galaxies.
