  The discrete symmetry $Z_4$ in the standard model (SM) with three
right-handed neutrinos is free from the Dai-Freed anomaly. Motivated by this
$Z_4$ symmetry, we constructed a topological inflation model consistent with
all known constraints and observations. However, we assumed a specific inflaton
potential in the previous work. In this paper we extend the inflaton potential
in a more general form allowed by the discrete $Z_4$ gauge symmetry and show
that consistent hilltop inflation is realized. We find that the Hubble
parameter $H_\mathrm{inf}$ can be smaller than $\simeq 10^{9}$ GeV so that the
isocurvature fluctuations of the axion dark matter are sufficiently suppressed.
Furthermore, the running of the spectral index can be as large as $dn_s/\ln k
\simeq 0.0018$ which will be tested in future CMB observations. Since this
discrete $Z_4$ acts on the SM, the inflaton can couple to pairs of the
right-handed neutrinos and hence the reheating temperature can be high as $\sim
10^{10}$ GeV, producing the cosmic baryon asymmetry naturally through the
thermal leptogenesis.
