  A generalization of inflationary $\alpha$-attractor models (``polynomial
$\alpha$-attractor'') was recently proposed by Kallosh and Linde, in which the
potential involves logarithmic functions of the inflaton so that the derivative
of the potential but not potential itself has a singularity. We find that the
models can lead to viable inflationary observables even without the pole in the
kinetic term. Also, the generalization with a pole order other than two does
not significantly change the functional form of the potential. This allows a
systematic analysis of the predictions of this class of models. Our models
predict larger spectral index $n_s$ and tensor-to-scalar ratio $r$ than in the
polynomial $\alpha$-attractor: typically, $n_s$ around 0.97--0.98 and $r$
observable by LiteBIRD. Taking advantage of the relatively large $n_s$, we
discuss the modification of the potential to produce primordial black holes as
the whole dark matter and gravitational waves induced by curvature
perturbations detectable by LISA and BBO/DECIGO, while keeping $n_s$ in
agreement with the Planck/BICEP/Keck data.
