Context. Co-orbital objects, also known as Trojans, are celestial bodies that share a 1:1 mean motion resonance orbit with a planet. Many planets in our Solar System hold Trojans, yet none has been confirmed in exoplanetary systems. While theoretical models suggest that hot Jupiters are unlikely to retain co-orbitals, their high transit cadence and deep transits make them accessible targets for constraining the existence of massive companions using transit timing variations (TTVs). Aims. The Exoplanet Ephemerides Change Observations (ExoEcho) project aims to study TTVs in exoplanetary systems using high-precision photometry from space- and ground-based telescopes. As the third paper in the ExoEcho project, we investigate the potential existence of exotrojans in hot-Jupiter systems through a TTV analysis of TESS observations. Methods. In this work, we analyzed TESS photometry for 260 confirmed hot Jupiters with published radial-velocity-based (RV-based) mass measurements to search for TTV signals compatible with Trojan companions. We derived transit times and assessed the compatibility of TTV residuals with co-orbital models through N-body simulation using the REBOUND N-body code. Accounting for the physical degeneracy between Trojan mass and libration amplitude, we placed upper mass limits on possible companions for typical libration amplitudes. Results. For a typical libration amplitude of 15°, we rule out exotrojans more massive than 1 Earth mass in 130 systems (∼50% of our sample). A more conservative χ2 analysis incorporating observational uncertainties places this limit at 3 Earth masses. Additionally, we used the stability boundaries of the 1:1 resonance to exclude dynamically unstable configurations, ensuring our derived mass limits fall within the stable regime. Conclusions. Our results provide stringent constraints on co-orbital formation in short-period systems and establish a framework for future high-precision missions such as the PLAnetary Transits and Oscillations of stars (PLATO) or Earth 2.0 (ET) mission.