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.

18 pages, 9 figures, 6 tables, Accepted for publication in Astronomy & Astrophysics