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 to a large peak (enhancement) in the power spectrum of scalar perturbations, which results in an efficient PBH formation. Both inflation and PBH formation are generic in our models, while these 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 can be realized only 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 the PBH-induced gravitational waves.