In todays global highly competitive markets, competition happens among supply chains instead of companies, as the members of supply chains. So, the partners of the chains seek to apply efficient coordinating strategies like discount, return, refund, buyback, or the other coordinating policies to abate the operation costs of the chains and subsequently increase market shares. Hence, because of the importance and application of these strategies in the current non-exclusive markets, in this study, we introduce different composite coordinating strategies to enhance the coordination of the supply chains. Here, we consider two competing supply chains where both chains launch the same product under different brands to the market by applying different composite coordinating strategies. Each supply chain comprises one manufacturer and a group of non-competing retailers where the manufacturer receives raw materials from an outside supplier and transforms them into a finished product; then, the products are sold to the retailers to satisfy the demands of market. In the first chain, a composite (QFF) policy, which is the combination of quantity and freight discount, as well as free shipping quantity policies, are considered between upstream and downstream members while in the second one, different composite polices are considered between upstream and downstream members such that the supplier offers a composite policy, as the first chain, to the manufacturer and the manufacturer proposes a composite (QPR) policy, which is the combination of quantity discount and partial-refund customer return policies, to the retailers. The main objective of the paper is to determine the optimal selling prices and the order quantities of the manufacturer and the retailers in each chain in presence of different composite coordinating strategies. A Stackelberg game-theoretic approach is employed between the members of each chain where the manufacturer is a follower and the retailers are leaders. The concavity of profit functions is proved. Finally, the applicability of the models is justified by presented numerical examples. Moreover, the effects of these strategies on the decisions of the chains partners are examined.