We present high-resolution multiline spectropolarimetric observations of a quiet-Sun network element obtained with the SUNRISE III Chromospheric Infrared SpectroPolarimeter. The observations combine photospheric, upper-photospheric, and chromospheric diagnostics at a spatial resolution and polarimetric sensitivity that allow the transverse magnetic structure of the network boundary to be examined directly. We find that the strongest linear polarization is concentrated in a narrow ridge around the edge of the magnetic element, cospatial with enhanced transverse magnetic field inferred from multiline inversions. The magnetic azimuth exhibits a coherent, predominantly radial organization around a more vertical core, consistent with an expanding magnetic canopy. An azimuth proxy derived directly from the observed Fe I and K I linear polarization reproduces the same large-scale organization, showing that this structure is encoded in the Stokes profiles rather than imposed by the inversion. Response functions and a MURaM-based forward-synthesis test indicate that the Fe I 8468 Å linear polarization is sensitive to magnetic azimuth in the upper photosphere, with the closest proxy agreement occurring near logτ≈-3 . We find no evidence for strong azimuthal shear between the Fe- and K-sensitive diagnostics. At the network boundary, we also identify localized parasitic-polarity patches associated with complex, multilobed Stokes V profiles and one case in which the Stokes V polarity reverses between photospheric Fe I and chromospheric Ca II lines. These results demonstrate that quiet-Sun network boundaries contain organized upper-photospheric canopy fields together with a small-scale mixed-polarity structure, providing new constraints on the three-dimensional magnetic structure of network elements.