Skipper Charge-Coupled Devices (Skipper-CCDs) are a leading technology in the search for sub-GeV dark matter and coherent elastic neutrino-nucleus scattering. A key background for rare-event searches with these detectors arises from "spurious charge" -- single-electron events generated when charges are transferred through the active region to the serial register, and across the serial register to the readout stage. We present a characterization of spurious charge in both the active region and the serial register of SENSEI Skipper-CCDs, and show that, in a well-shielded low-background environment, the dominant contribution originates in the serial register during Skipper readout, when horizontal clocks are held at constant voltage between pixel transfers. Motivated by this finding, we develop a "tri-level" clocking scheme in which the held-low phase is raised to an intermediate voltage during readout to suppress trap-mediated charge generation. Using the SENSEI detector near the MINOS cavern, we measure a serial-register single-electron density of $(2.9 \pm 0.1) \times 10^{-5}$ electrons/pixel/image under standard SENSEI readout conditions, reduced to $(4.0 \pm 0.4) \times 10^{-6}$ electrons/pixel/image with tri-level clocking -- a factor of $\sim$7 improvement. This technique offers a promising path to lower backgrounds in current and future Skipper-CCD experiments.