Almost every known hot Jupiter is “lonely,” meaning very few companion planets have been found to exist nearby within an orbit of ~200 days. The existence of nearby planets, or lack thereof, can place constraints and provide clues as to the mechanism through which these hot Jupiters formed. We present the results of an independent, uniform search for additional planets around all stars with confirmed hot Jupiters observed by the Transiting Exoplanet Survey Satellite (TESS) in its Cycle 1 survey of the southern ecliptic hemisphere. Our search comprises 194 total systems with confirmed hot Jupiters, most of which were confirmed prior to the start of TESS science operations in July 2018. The Transit Least Squares (TLS) algorithm was utilized to search for periodic signals that may have been missed by other planet search pipelines. Our search yielded no new statistically-validated planetary candidates. A lack of planet candidates nearby hot Jupiters in the TESS data supports results from previous transit searches of each individual system, now down to the photometric precision of TESS. This is consistent with expectations from a high eccentricity migration formation scenario, but additional formation indicators are needed for definitive confirmation. We also injected transit signals into the light curves of the hot Jupiter sample to probe the detectable parameter space of potential planet signals using our TLS-based pipeline. We recover all injected planets with a Rp > 2.5 R⊕ and host star with TESS mag < 9.5 and find the recovery rate depends strongly on the magnitude of the host star, Rp, and orbital period outside of this range. This study demonstrates how TESS uniquely enables comprehensive searches for nearby planetary companions to nearly all the known hot Jupiters.