We briefly review known results on upper bounds for the minimal domination number $γ_n$ of a hypercube of dimension $n$, then present a new method for constructing dominating sets. Write $n =2^{\hat{n}}-1 +{\check{n}}$ with $0\leq {\check{n}}<2^{\hat{n}}$. Our construction applies to all $n$ lying within the expanding wedge $θ({\hat{n}}) \leq {\check{n}} < 2^{\hat{n}}$, where $θ$ is a specific, easily computable function with the asymptotic property $θ(a) \sim 2^{a/2}$. For all $n$ within the smaller wedge $θ({\hat{n}}) \leq {\check{n}} < 2^{\hat{n}-2}$, the resulting upper bound on $γ_n$ betters those previously known.