Order-of-addition (OofA) experiments have gained renewed attention in recent years, especially in regard to their design. For these experiments, the response is determined by the order in which components are added. A particularly useful design introduced in OofA experiments is the component orthogonal array (COA). The COA maintains pairwise balance between any two components while also ensuring each component appears equally often in each position. In this paper, we propose an efficient algorithm for constructing COAs which can be naturally split into blocks of Latin squares. These blocks can be run sequentially in a systematic order, potentially requiring fewer runs to identify optimal orderings, while also preserving good properties should the overall design be needed. We also show how to extend this construction method to create designs for any number of components.