We present connectable designs of superconducting undulators (SCU) and cryomodules (CM) based on previous SCU and CM designs at Argonne National Lab. The new SCU and CM designs will allow us to connect one CM to the next to form a contiguous line of SCUs with no breaks between the cryomodules. The SCU design will have correctors and phase shifters integrated into the main SCU magnet core, as well as external corrector magnets for trajectory corrections. There will also be a cryogenic magnetic quadrupole and a cold RF beam position monitor (BPM) integrated in the SCU CM. In addition to providing the usual FODO transverse focusing, the quadrupole and BPM will be used for the beam-based alignment technique that is critical for X-ray FEL operation. In this paper, we will present the conceptual design of the new SCU CM as well as results of FEL simulations using the SCUs as afterburners for the LCLS hard X-ray undulators. Proceedings of the 5th North American Particle Accelerator Conference, NAPAC2022, Albuquerque, NM, USA
