  We study the parameter space of the effective (with two scalars) models of
cosmological inflation and primordial black hole (PBH) formation in the
modified $(R+R^2)$ supergravity. Our models describe double inflation, whose
first stage is driven by Starobinsky's scalaron coming from the $R^2$ gravity,
and whose second stage is driven by another scalar belonging to the
supergravity multiplet. The ultra-slow-roll regime between the two stages leads
a large peak (enhancement) in the power spectrum of scalar perturbations, which
results in efficient PBH formation. Both inflation and PBH formation are
generic in our models, while those PBH can account for a significant part or
the whole of dark matter. Some of the earlier proposed models in the same class
are in tension (over $3\sigma$) with the observed value of the scalar tilt
$n_s$, so that we study more general models with more parameters, and
investigate the dependence of the cosmological tilts $(n_s,r)$ and the scalar
power spectrum enhancement upon the parameters. The PBH masses and their
density fraction (as part of dark matter) are also calculated. A good agreement
(between $2\sigma$ and $3\sigma$) with the observed value of $n_s$ requires
fine tuning of the parameters, and it is only realized in the so-called
$\delta$-models. Our models offer the (super)gravitational origin of inflation,
PBH and dark matter together, and may be confirmed or falsified by future
precision measurements of the cosmic microwave background radiation and
PBH-induced gravitational waves.
