In this paper, we explore an efficient uncoupled unsourced random access (UURA) scheme for 6G massive communication. UURA is a typical framework of unsourced random access that addresses the problems of codeword detection and message stitching, without the use of check bits. Firstly, we establish a framework for UURA, allowing for immediate decoding of sub-messages upon arrival. Thus, the processing delay is effectively reduced due to the decreasing waiting time. Next, we propose an integrated decoding algorithm for sub-messages by leveraging matrix information geometry (MIG) theory. Specifically, MIG is applied to measure the feature similarities of codewords belonging to the same user equipment, and thus sub-message can be stitched once it is received. This enables the timely recovery of a portion of the original message by simultaneously detecting and stitching codewords within the current sub-slot. Furthermore, we analyze the performance of the proposed integrated decoding-based UURA scheme in terms of computational complexity and convergence rate. Finally, we present extensive simulation results to validate the effectiveness of the proposed scheme in 6G wireless networks.