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

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 $x^{6}-2x^{4}-x^{3}-1=0$ .