We establish a refined version of a graph container lemma due to Galvin and discuss several applications related to the hard-core model on bipartite expander graphs. Given a graph $G$ and $λ>0$, the hard-core model on $G$ at activity $λ$ is the probability distribution $μ_{G,λ}$ on independent sets in $G$ given by $μ_{G,λ}(I)\propto λ^{|I|}$. As one of our main applications, we show that the hard-core model at activity $λ$ on the hypercube $Q_d$ exhibits a `structured phase' for $λ= Ω( \log^2 d/d^{1/2})$ in the following sense: in a typical sample from $μ_{Q_d,λ}$, most vertices are contained in one side of the bipartition of $Q_d$. This improves upon a result of Galvin which establishes the same for $λ=Ω(\log d/ d^{1/3})$. As another application, we establish a fully polynomial-time approximation scheme (FPTAS) for the hard-core model on a $d$-regular bipartite $α$-expander, with $α>0$ fixed, when $λ= Ω( \log^2 d/d^{1/2})$. This improves upon the bound $λ=Ω(\log d/ d^{1/4})$ due to the first author, Perkins and Potukuchi. We discuss similar improvements to results of Galvin-Tetali, Balogh-Garcia-Li and Kronenberg-Spinka.