Images transmitted through social networks generally undergo JPEG recompression, which degrades image quality and could disrupt embedded secret messages, making error-free extraction of such messages difficult. In this paper, we propose an INN-based robust JPEG steganography framework through cover coefficient selection, which selects appropriate embedding coefficients based on prior knowledge of JPEG compression and the characteristics of invertible neural networks (INNs). When subjected to JPEG recompression with quality factor (QF)=90 and QF=85 during transmission, the proposed method achieves extraction accuracy as high as 100%. Moreover, we introduce a novel learnable noise layer that incorporates rounding and truncation operations into network training, to mitigate the information loss caused by rounding and to limit pixel values of the stego image within the range of [0-255] as much as possible. When compared with existing INN-based methods, our method reduces runtime by 23% and memory usage by 82.14%.