High-quality images are prevalent on social media, presenting opportunities for covert communication through image steganography. However, platforms often recompress images to save on bandwidth and storage, corrupting hidden data. This paper proposes a novel robust steganography method for high fidelity JPEG images. The technique meticulously selects discrete cosine transform (DCT) coefficients and uses an adaptive embedding algorithm that minimally modifies those coefficients to encode secret messages. For the test cases examined, the proposed approach demonstrates the ability to reliably recover hidden information with 96% accuracy even after recompression from quality 95 to 70, significantly outperforming prior arts. The method also provides goodness in imperceptibility and statistical undetectability across evaluated payload sizes. The results on benchmark datasets indicate potential of the proposed method for practical steganography applications.