In this paper, we report on X-ray transition-edge sensor (TES) microcalorimeters optimized to have the best possible energy resolution for a limited energy range for the incoming X-rays, such as an energy resolution of 0.3 eV full width half maximum (FWHM) for energies up to ≈ 0.8 keV as is desirable for one of the Lynx X-ray Microcalorimeter subarrays. The test array we have fabricated has 60 × 60 sensors on a pitch of 50 μm , and has 46 × 46 μm 2 absorbers that are one micrometer thick. We have measured a spectral energy resolution of the same device using 3 eV photons delivered through an optical fiber. For the one-photon 3 eV line, we have obtained an energy resolution of 0.25 eV FWHM, which is consistent with the estimated performance based on the signal size and noise. Further measurements will determine how the energy resolution degrades with energy. Based upon measurements of the TES transition characteristics, it appears that this level of energy resolution should be achievable up to 0.5 keV, and the performance will then gradually degrade to the measured energy resolution of around 2.3 eV at 1.5 keV. In this paper, we describe the full design and characterization of this detector, and discuss the performance limits of pixels designs like this.