  A new (improved) model of inflation and primordial black hole (PBH) formation
is proposed by combining Starobinsky model of inflation,
Appleby-Battye-Starobinsky (ABS) model of dark energy and a quantum correction
in the modified $F(R)$ gravity. The energy scale parameter in the ABS model is
taken to be close to the inflationary scale, in order to describe double
inflation instead of dark energy. The quantum correction is given by the term
quartic in the spacetime scalar curvature $R$ with a negative coefficient
$(-\delta)$ in the $F(R)$ function. It is demonstrated that very good agreement
(within $1\sigma$) with current measurements of the cosmic microwave background
(CMB) radiation can be achieved by choosing the proper value of $\delta$, thus
solving the problem of low values of the tilt of CMB scalar perturbations in
the earlier proposed model in arXiv:2205.00603. A large (by the factor of
$10^7$ against CMB) enhancement in the power spectrum of scalar perturbations
is achieved by fine-tuning the parameters of the model. It is found by
numerical analysis that it can lead to formation of asteroid-size PBH with the
masses up to $10^{20}$ g, which may form dark matter in the current universe.
