Exhaustion of the curve graph via rigid expansions

For an orientable surface $S$ of finite topological type with genus $g \geq 3$, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph of $S$. The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.