The minimum achievable particle beam emittance in an electron accelerator depends strongly on the intrinsic emittance of the photocathode electron source. This is measurable as the mean longitudinal and transverse energy spreads in the photoemitted electron beam (MLE and MTE respectively); consequently, MLE and MTE are notable figures of merit for photocathodes used as electron sources in particle accelerators. The overall energy spread is defined by the sum of the MTE and the MLE, and the minimization of MTE is crucial to reduce emittance and thus generate a high-brightness electron beam. Reducing the electron beam emittance in an accelerator that drives a Free-Electron Laser (FEL) delivers a significant reduction in the saturation length for an x-ray FEL, thus reducing the machine's construction footprint and operating costs while increasing the x-ray beam brightness. The ability to measure the transverse energy distribution curve of photoelectrons emitted from a photocathode is a key enabler in photocathode research and development that has prompted the Accelerator Science and Technology Centre (ASTeC) at the STFC Daresbury Laboratory to develop the Transverse Energy Spread Spectrometer to make these crucial measurements. We present details of the design for the upgraded TESS instrument with measured data for copper (100), (110), and (111) single-crystal photocathodes illuminated at UV wavelengths around 266 nm.