Computed tomography (CT) using synchrotron radiation is a powerful technique that, compared with laboratory CT techniques, boosts high spatial and temporal resolution while also providing access to a range of contrast-formation mechanisms. The acquired projection data are typically processed by a computational pipeline composed of multiple stages. Artifacts introduced during data acquisition can propagate through the pipeline and degrade image quality in the reconstructed images. Recently, deep learning has shown significant promise in enhancing image quality for images representing scientific data. This success has driven increasing adoption of deep learning techniques in CT imaging. Various approaches have been proposed to incorporate deep learning into computational pipelines, but each has limitations in addressing artifacts effectively and efficiently in synchrotron CT, either in properly addressing the specific artifacts or in computational efficiency. Recognizing these challenges, we introduce a novel method that incorporates separate deep learning models at each stage of the tomography pipeline - projection, sinogram and reconstruction - to address specific artifacts locally in a data-driven way. Our approach includes bypass connections that feed both the outputs from previous stages and raw data to subsequent stages, minimizing the risk of error propagation. Extensive evaluations on both simulated and real-world datasets illustrate that our approach effectively reduces artifacts and outperforms comparison methods.