  We present constraints on the extended Starobinsky and Weyl gravity model of
inflation using updated available observational data. The data includes cosmic
microwave background (CMB) anisotropy measurements from Planck and BICEP/Keck
2018 (BK18), as well as large-scale structure data encompassing cosmic shear
and galaxy autocorrelation and cross-correlation functions measurements from
Dark Energy Survey (DES), baryonic acoustic oscillation (BAO) measurements from
6dF, MGS and BOSS, and distance measurements from supernovae type Ia from
Pantheon+ samples. By introducing a single additional parameter, each model
extends the Starobinsky model to encompass larger region of parameter space
while remaining consistent with all observational data. Our findings
demonstrate that the inclusion of higher-order terms loosen the constraint on
the upper bound of $e$-folding number $N_{\rm e}$ due to the presence of small
additional parameter. The maximum limit on $N_{\rm e}$ could be refined by
considering the reheating process to $N_{\rm e}<55-59$ for $k_{*}=0.002-0.05\
{\rm Mpc}^{-1}$. These models extend viable range of tensor-to-scalar ratio
($r$) to very small value $r<0.002$ in contrast to the original $R^2$
Starobinsky model. In addition, our results continue to emphasize the tension
in $H_0$ and $S_8$ between early-time CMB measurements and late-time
large-scale structure observations.
