Context. Compact hierarchical triple (CHT) systems, where a tertiary component orbits an inner binary, provide critical insights into stellar formation and evolution. Despite their importance, the detection of such systems, especially compact ones, remains challenging due to the complexity of their orbital dynamics and the limitations of traditional observational methods. Aims. This study aims to identify new CHT star systems among Gaia astrometric binaries and accelerated solutions by analysing the radial velocity (RV) amplitude of these systems, thereby improving our understanding of stellar hierarchies. Methods. We selected a sample of bright astrometric binaries and accelerated solutions from the Gaia DR3 Non-Single Stars catalogue. The RV peak-to-peak amplitude was used as an estimator, and we applied a new method to detect potential triple systems by comparing the RV-based semi-amplitude with the astrometric semi-amplitude. We used available binary and triple star catalogues to identify and validate candidates, with a subset confirmed through further examination of the RV and astrometric data. Results. Our analysis resulted in the discovery of 956 CHT candidates among the orbital sources as well as another 3115 probable close binary sources in stars with accelerated solutions. Exploring the inclination, orbital period, and eccentricity of the outer companion in these CHT systems provides strong evidence of mutual orbit alignment, as well as a preference towards moderate outer eccentricities. Conclusions. Our novel approach has proven effective in identifying potential triple systems, thereby increasing their number in the catalogues. Our findings emphasise the importance of combined astrometric and RV data analysis in the study of multiple star systems.