For a graph $G$, let $Π(G)$ denote the set of all potential maximal cliques of $G$. For each subset $Π$ of $Π(G)$, let $\tw(G, Π)$ denote the smallest $k$ such that there is a tree-decomposition of $G$ of width $k$ whose bags all belong to $Π$. Bouchitté and Todinca observed in 2001 that $\tw(G, Π(G))$ is exactly the treewidth of $G$ and developed a dynamic programming algorithm to compute it. Indeed, their algorithm can readily be applied to an arbitrary non-empty subset $Π$ of $Π(G)$ and computes $\tw(G, Π)$, or reports that it is undefined, in time $|Π||V(G)|^{O(1)}$. This efficient tool for computing $\tw(G, Π)$ allows us to conceive of an iterative improvement procedure for treewidth upper bounds which maintains, as the current solution, a set of potential maximal cliques rather than a tree-decomposition. We design and implement an algorithm along this approach. Experiments show that our algorithm vastly outperforms previously implemented heuristic algorithms for treewidth.