  We present the results of a 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 184 total planetary systems with confirmed hot Jupiters with
$R_{p}$ > 8$R_\oplus$ and orbital period < 10 days. The Transit Least Squares
(TLS) algorithm was utilized to search for periodic signals that may have been
missed by other planet search pipelines. While we recovered 169 of these
confirmed hot Jupiters, our search yielded no new statistically-validated
planetary candidates in the parameter space searched (P < 14 days). 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 injected transit signals into the
light curves of the hot Jupiter sample to probe the pipeline's sensitivity to
the target parameter space, finding a dependence proportional to
$R_{p}^{2.32}P^{-0.88}$ for planets within 0.3$\leq$$R_{p}$$\leq$4 $R_\oplus$
and 1$\leq$$P$$\leq$14 days. A statistical analysis accounting for this
sensitivity provides a median and $90\%$ confidence interval of
$7.3\substack{+15.2 \\ -7.3}\%$ for the rate of hot Jupiters with nearby
companions in this target parameter space. This study demonstrates how TESS
uniquely enables comprehensive searches for nearby planetary companions to
nearly all the known hot Jupiters.
