How small can a set of vertices in the $n$-dimensional hypercube $Q_n$ be if it meets every copy of $Q_d$? The asymptotic density of such a set (for $d$ fixed and $n$ large) is denoted by $γ_d$. It is easy to see that $γ_d \leq 1/(d+1)$, and it is known that $γ_d=1/(d+1)$ for $d \leq 2$, but it was recently shown that $γ_d < 1/(d+1)$ for $d \geq 8$. In this paper we show that the latter phenomenon also holds for $d=7$ and $d=6$.