\phi$ decreases as $ a^{-3/4}$. Thus, the initial condition for the new inflation is estimated as $${ { \phi}_{\mathrm{ini}}\simeq-}\frac{{ v}^{3}}{{\mu}^{3}}{\sigma}_{ c}.$$ Taking into account this oscillatory phase, and assuming the ordinary thermal history after the reheating, we estimate the total number of $e$-folds $ N_{\mathrm{tot}}$ as $$\label{eq:2:ntot}
 N_{\mathrm{tot}}
=  N_{H} + N_{\mathrm{new}}
=  57.6 + \frac{2}{3}\ln\mu + \frac{1}{3}\ln \left(\frac{T_{\mathrm{R}}}{10^9\mathrm{GeV}}\right),$$ where we set the reheating temperature as $T_{\mathrm{R}}$.

Parameter search for $\sigma_{i}$
---------------------------------

In this subsection, we search for the allowed region of $\sigma_{i}$ at each $n_s$. At the pivot scale $k_0$, the spectral index $n_s$ and the curvature perturbation $\cal R$ are written in terms of $\sigma_{i}$, the energy scale of the smooth hybrid inflation $\mu$ and the cut-off scale $M$. Then, using the Planck constraint ${\cal R}\simeq4.7\times10^{-5}$  [@Ade:2013zuv], we can write down $\mu$ and $M$ in terms of $n_s$ and $\sigma_{i}$ as $$\label{eq:mu}
\mu 
= 7.2\times10^{-3}~
\sqrt{
8~\sigma_{i}^3-\left(n_s-1\right)\sigma_{i}
},$$ $$\label{eq:M}
M
=
81~\sigma_{i}^3
\sqrt{\frac{
3~\sigma_{i}^2-\left(n_s-1\right)
}{
8~\sigma_{i}^3-\left(n_s-1\right)\sigma_{i}
}}.$$ The value of the inflaton $\sigma_{c}$, at which the smooth hybrid inflation ends, is also written in terms of $\sigma_{i}$ and $n_s$ as $$\sigma_{c}
=
0.89~\sigma_{i}
\left[3~\sigma_{i}^2-\left(n_s-1\right)\right]^{1/6}.$$ The $e$-folding number of the smooth hybrid inflation $N_{\mathrm{H}}$ and that of the new 