  We continue the study, initiated in [Giacomin and Lacoin, JEMS 2018], of the
localization transition of a lattice free field $\phi=(\phi(x))_{x \in Z^d}$,
$d\ge 3$, in presence of a quenched disordered substrate. The presence of the
substrate affects the interface at the spatial sites in which the interface
height is close to zero. This corresponds to the Hamiltonian $$ \sum_{x\in Z^d
}(\beta \omega_x+h)\delta_x,$$ where $\delta_x=1_{[-1,1]}(\phi(x))$, and
$(\omega_x)_{x\in Z^d}$ is an IID centered field. A transition takes place when
the average pinning potential $h$ goes past a threshold $h_c(\beta)$: from a
delocalized phase $h<h_c(\beta)$, where the field is macroscopically repelled
by the substrate, to a localized one $h>h_c(\beta)$ where the field sticks to
the substrate. In [Giacomin and Lacoin, JEMS 2018] the critical value of $h$ is
identified and it coincides, up to the sign, with the $\log$-Laplace transform
of $\omega=\omega_x$, that is $-h_c(\beta)=\lambda(\beta):=\log
E[e^{\beta\omega}]$. Here we obtain the sharp critical behavior of the free
energy approaching criticality: $$\lim_{u\searrow 0} \frac{
F(\beta,h_c(\beta)+u)}{u^2}= \frac{1}{2\, \textrm{Var}\left(e^{\beta
\omega-\lambda(\beta)}\right)}.$$ Moreover, we give a precise description of
the trajectories of the field in the same regime: the absolute value of the
field is $\sqrt{2\sigma_d^2\vert\log(h-h_c(\beta))\vert}$ to leading order when
$h\searrow h_c(\beta)$ except on a vanishing fraction of sites ($\sigma_d^2$ is
the single site variance of the free field).
