We present StHorn, a novel technique for solving the satisfiability problem of CHCs, which works lazily and incrementally and is guided by the structure of the set of CHCs. Our technique is driven by the idea that a set of CHCs can be solved in parts, making it an easier problem for the CHC-solver. Furthermore, solving a set of CHCs can benefit from an interpretation revealed by the solver for its subsets. Our technique is lazy in that it gradually extends the set of checked CHCs, as needed. It is incremental in the way it constructs a solution by using satisfying interpretations obtained for previously checked subsets. In order to capture the structure of the problem, we define an induced CHC hypergraph that precisely corresponds to the set of CHCs. The paths in this graph are explored and used to select the clauses to be solved. We implemented StHorn on top of two CHC-solvers, Spacer and Eldarica. Our evaluation shows that StHorn complements both tools and can solve instances that cannot be solved by the other tools. We conclude that StHorn can improve upon the state-of-the-art in CHC solving.