As a cutting-edge technique of the sixth generation mobile communication, reconfigurable intelligent surface (RIS) is compatible with many existing technologies very well. In the existing literature, the inter-RIS reflecting signals are usually ignored for simplification. In this letter, we consider a double cooperative-RIS-assisted uplink non-orthogonal multiple access (NOMA) system with inter-RIS reflections. Then, we formulate an optimization problem of minimizing the total transmission power for cooperative-RIS-assisted NOMA uplinks. In order to simplify the original problem, we derive the relationship between the optimal transmission power and the phase shifts. Besides, a block coordinate descent (BCD) method is proposed to optimize the phase shifts of the two RISs alternatively. The simulation results show that the proposed optimization algorithm achieves better performance than the existing ones.