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The Optically Segmented Single Volume Scatter Camera (OS-SVSC) aims to image neutron sources for nuclear non-proliferation applications using the kinematic reconstruction of elastic double-scatter events. We report on the design, construction, and calibration of one module of a new prototype. The module includes 16 EJ-204 organic plastic scintillating bars individually wrapped in Teflon tape, each measuring 0.5 cm$\times$0.5 cm$\times$20 cm. The scintillator array is coupled to two custom Silicon Photomultiplier (SiPM) boards consisting of a 2$\times$8 array of SensL J-Series-60035 Silicon Photomultipliers, which are read out by a custom 16 channel DRS-4 based digitizer board. The electrical crosstalk between SiPMs within the electronics chain is measured as 0.76% $\pm$ 0.11% among all 16 channels. We report the detector response of one module including interaction position, time, and energy, using two different optical coupling materials: EJ-560 silicone rubber optical coupling pads and EJ-550 optical coupling grease. We present results in terms of the overall mean and standard deviation of the z-position reconstruction and interaction time resolutions for all 16 bars in the module. We observed the z-position resolution for gamma interactions in the 0.3 MeVee to 0.4 MeVee range to be 2.24 cm$\pm$1.10 cm and 1.45 cm$\pm$0.19 cm for silicone optical coupling pad and optical grease, respectively. The observed interaction time resolution is 265 ps$\pm$29 ps and 235 ps$\pm$10 ps for silicone optical coupling pad and optical grease, respectively., Comment: 11 pages, 21 figures. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible