Given a weighted hypergraph $\mathcal{H}(V, \mathcal{E} \subseteq 2^V, w)$, the approximate $k$-cover problem seeks for a size-$k$ subset of $V$ that has the maximum weighted coverage by \emph{sampling only a few hyperedges} in $\mathcal{E}$. The problem has emerged from several network analysis applications including viral marketing, centrality maximization, and landmark selection. Despite many efforts, even the best approaches require $O(k n \log n)$ space complexities, thus, cannot scale to, nowadays, humongous networks without sacrificing formal guarantees. In this paper, we propose BCA, a family of algorithms for approximate $k$-cover that can find $(1-\frac{1}{e} -ε)$-approximation solutions within an \emph{$O(ε^{-2}n \log n)$ space}. That is a factor $k$ reduction on space comparing to the state-of-the-art approaches with the same guarantee. We further make BCA more efficient and robust on real-world instances by introducing a novel adaptive sampling scheme, termed DTA.