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This paper identifies spectral lines from X-ray to infrared wavelengths which are optimally suited to measuring vector magnetic fields as high as possible in the solar atmosphere. Instrumental and Earth's atmospheric properties, as well as solar abundances, atmospheric properties and elementary atomic physics are considered without bias towards particular wavelengths or diagnostic techniques. While narrowly-focused investigations of individual lines have been reported in detail, no assessment of the comparative merits of all lines has ever been published. Although in the UV, on balance the Mg+ h and k lines near 2800 Angstroms are optimally suited to polarimetry of plasma near the base of the solar corona. This result was unanticipated, given that longer-wavelength lines offer greater sensitivity to the Zeeman effect. While these lines sample optical depths photosphere to the coronal base, we argue that cores of multiple spectral lines provide a far more discriminating probe of magnetic structure as a function of optical depth than the core and inner wings of a strong line. Thus, together with many chromospheric lines of Fe+ between 2585 and the h line at 2803 Angstrom, this UV region promises new discoveries concerning how the magnetic fields emerge, heat, and accelerate plasma as they battle to dominate the force and energy balance within the poorly-understood chromosphere.