The Tucana-Horologium association (Tuc-Hor) is a 40Myr old moving group in the southern sky. In this work, we measure the rotation periods of 313 Tuc-Hor objects with TESS light curves derived from TESS full-frame images and membership lists driven by Gaia EDR3 kinematics and known youth indicators. We recover a period for 81.4% of the sample and report 255 rotation periods for Tuc-Hor objects. From these objects we identify 11 candidate binaries based on multiple periodic signals or outlier Gaia DR2 and EDR3 renormalized unit weight error values. We also identify three new complex rotators (rapidly rotating M dwarf objects with intricate light-curve morphology) within our sample. Along with the six previously known complex rotators that belong to Tuc-Hor, we compare their light-curve morphology between TESS Cycle 1 and Cycle 3 and find that they change substantially. Furthermore, we provide context for the entire Tuc-Hor rotation sample by describing the rotation period distributions alongside other youth indicators such as H{alpha} and Li equivalent width, as well as near-ultraviolet and X-ray flux. We find that measuring rotation periods with TESS is a fast and effective means to confirm members in young moving groups.