Let $q\in (1,2)$. A $q$-expansion of a number $x$ in $[0,1/(q-1)]$ is a sequence $({\it\delta}_{i})_{i=1}^{\infty }\in \{0,1\}^{\mathbb{N}}$ satisfying
$$\begin{eqnarray}x=\mathop{\sum }_{i=1}^{\infty }\frac{{\it\delta}_{i}}{q^{i}}.\end{eqnarray}$$ Let
${\mathcal{B}}_{\aleph _{0}}$ denote the set of
$q$ for which there exists
$x$ with a countable number of
$q$-expansions, and let
${\mathcal{B}}_{1,\aleph _{0}}$ denote the set of
$q$ for which
$1$ has a countable number of
$q$-expansions. In Erdős
et al [On the uniqueness of the expansions
$1=\sum _{i=1}^{\infty }q^{-n_{i}}$.
Acta Math. Hungar.58 (1991), 333–342] it was shown that
$\min {\mathcal{B}}_{\aleph _{0}}=\min {\mathcal{B}}_{1,\aleph _{0}}=(1+\sqrt{5})/2$, and in S. Baker [On small bases which admit countably many expansions.
J. Number Theory147 (2015), 515–532] it was shown that
${\mathcal{B}}_{\aleph _{0}}\cap ((1+\sqrt{5})/2,q_{1}]=\{q_{1}\}$, where
$q_{1}\,({\approx}1.64541)$ is the positive root of
$x^{6}-x^{4}-x^{3}-2x^{2}-x-1=0$. In this paper we show that the second smallest point of
${\mathcal{B}}_{1,\aleph _{0}}$ is
$q_{3}\,({\approx}1.68042)$, the positive root of
$x^{5}-x^{4}-x^{3}-x+1=0$. En route to proving this result, we show that
${\mathcal{B}}_{\aleph _{0}}\cap (q_{1},q_{3}]=\{q_{2},q_{3}\}$, where
$q_{2}\,({\approx}1.65462)$ is the positive root of
$x^{6}-2x^{4}-x^{3}-1=0$.