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We present the design and science goals of SPT-3G+, a new camera for the South Pole Telescope, which will consist of a dense array of 34100 kinetic inductance detectors measuring the cosmic microwave background (CMB) at 220 GHz, 285 GHz, and 345 GHz. The SPT-3G+ dataset will enable new constraints on the process of reionization, including measurements of the patchy kinematic Sunyaev-Zeldovich effect and improved constraints on the optical depth due to reionization. At the same time, it will serve as a pathfinder for the detection of Rayleigh scattering, which could allow future CMB surveys to constrain cosmological parameters better than from the primary CMB alone. In addition, the combined, multi-band SPT-3G and SPT-3G+ survey data will have several synergies that enhance the original SPT-3G survey, including: extending the redshift-reach of SZ cluster surveys to $z > 2$; understanding the relationship between magnetic fields and star formation in our Galaxy; improved characterization of the impact of dust on inflationary B-mode searches; and characterizing astrophysical transients at the boundary between mm and sub-mm wavelengths. Finally, the modular design of the SPT-3G+ camera allows it to serve as an on-sky demonstrator for new detector technologies employing microwave readout, such as the on-chip spectrometers that we expect to deploy during the SPT-3G+ survey. In this paper, we describe the science goals of the project and the key technology developments that enable its powerful yet compact design.