  In this work we focus on the evolution of the linear perturbations in the
novel hybrid metric-Palatini theory achieved by adding a $f(\mathcal{R})$
function to the gravitational action. Working in the Jordan frame, we derive
the full set of linearized evolution equations for the perturbed potentials and
present them in the Newtonian and synchronous gauges. We also derive the
Poisson equation, and perform the evolution of the lensing potential,
$\Phi_{+}$, for a model with a background evolution indistinguishable from
$\Lambda$CDM. In order to do so, we introduce a designer approach that allows
to retrieve a family of functions $f(\mathcal{R})$ for which the effective
equation of state is exactly $w_{\textrm{eff}} = -1$. We conclude, for this
particular model, that the main deviations from standard General Relativity and
the Cosmological Constant model arise in the distant past, with an oscillatory
signature in the ratio between the Newtonian potentials, $\Phi$ and $\Psi$.
