the inflaton $ \phi$ is smaller than $ { \phi}_{c}$ at which slow-roll condition is broken, $|{ \eta}|=\frac{1}{2}|\frac{V_{N}''}{V_{N}}|\sim1$, and $\phi_c$ is given by $${ {\phi}_{c} =}
\sqrt{\frac{{ v}^{ 2}}{{ 6}{ g}} {\left(1-C_{N}\right)}}.$$ Then, the number of e-folds $ N_{\mathrm{new}}$ is written as $$N_{\mathrm{new}} \simeq
\frac{1}{C_{N}}\ln\frac{{\phi}_{d}}{{ \phi}_{i}} 
+ \frac{1-4C_{N}}{2C_{N}\left(1-C_{N}\right)}.$$ Here we have assumed that $ C_{N}\leq \frac{1}{4}$.

Oscillatory phase
-----------------

Here, let us consider the oscillatory phase from the end of the smooth hybrid inflation to the start of the new inflation.

After the end of the smooth hybrid inflation, $ \sigma$ and $ \psi$ roll down toward ${ \sigma}_{\mathrm{min}} = 0$ and $ { \psi}_{\mathrm{min}} = 2\sqrt{{ \mu}M}$ where the potential has the minimum. Around the potential minimum, their effective masses $ m_{ \sigma}$ and $ m_{ \psi}$ are given by $$\begin{array}{l}
 m_{ \sigma} = \sqrt{\frac{8{\mu}^{3}}{M}}\gg H,\\[0.5\intextsep]
 m _{ \psi} = \sqrt{\frac{8{\mu}^{3}}{M} + 16{ \mu}^{4}}\gg H.
\end{array}$$ Thus, $ \sigma$ and $ \psi$ oscillate around their respective minima and the total energy density decreases as $ a^{-3}$, where $a$ is the scale factor. Eventually, the false vacuum energy of the new inflation $ v^{4}$ dominates the total energy density of the universe, and then new inflation starts. During this oscillatory 