  Aims. We introduce a novel way to identify new compact hierarchical triple
stars by exploiting the huge potential of Gaia DR3 and also its future data
releases. We aim to increase the current number of compact hierarchical triples
significantly. Methods. We utilize several eclipsing binary catalogs from
different sky surveys totaling more than 1 million targets for which we search
for Gaia DR3 Non-single Star orbital solutions with periods substantially
longer than the eclipsing periods of the binaries. Those solutions in most
cases should belong to outer orbits of tertiary stars in those systems. We also
try to validate some of our best-suited candidates using TESS eclipse timing
variations. Results. We find 403 objects with suitable Gaia orbital solutions
of which 27 are already known triple systems. This makes 376 newly identified
hierarchical triple system candidates in our sample. We analyze the cumulative
probability distribution of the outer orbit eccentricities and find that it is
very similar to the ones found by earlier studies based on the observations of
the Kepler and OGLE missions. We found measurable non-linear eclipse timing
variations or third-body eclipses in the TESS data for 192 objects which we
also consider to be confirmed candidates. Out of these, we construct analytical
light-travel time effect models for the eclipse timing variations of 22 objects
with well-sampled TESS observations. We compare the outer orbital parameters
from our solutions with the ones from the Gaia solutions and find that the most
reliable orbital parameter is the orbital period, while the values of the other
parameters should be used with caution.
