Context. Hypervelocity stars are unique objects that move through the Milky Way at speeds exceeding the local escape velocity. They provide valuable insights into the gravitational potential of the Galaxy and the properties of its central supermassive black hole. The advent of Gaia DR3 offers an unprecedented astrometric precision that enables the discovery of new hypervelocity stars and facilitates their characterization. Aims. This study seeks to identify and characterize hypervelocity star candidates using Gaia DR3 data. We focus on stars without radial velocity measurements. Our goal was to estimate the total velocities of these stars and establish their origin within the Galactic framework, if possible. Methods. We applied strict selection criteria to Gaia DR3 data by focusing on sources with low parallax uncertainties and high astrometric fidelity. The distributions of the total velocities in the Galactic rest frame were derived and used to identify candidates. Spectroscopic follow-up with VLT/FORS2 provided radial velocity measurements for a subset of these candidates. We evaluated the probabilities of stars that exceeded local escape velocities under different Galactic potential models and traced their past orbits to identify possible origins. Results. From Gaia DR3, we identified 149 hypervelocity star candidates with probabilities Pesc ≥ 50% of exceeding local escape velocities. Our follow-up spectroscopy for 23 of these sources confirmed that the selected targets travel at high velocities, and many appear to escape the Galaxy, depending on the adopted Galactic potential. We found that except for one target with a minimum distance of ~1 kpc within uncertainties, none of the targets seems to have originated at the Galactic center. On the other hand, our analysis suggests that nearly one-third of the stars may have an extragalactic origin. These findings highlight the need for more precise astrometric and spectroscopic data to conclusively determine the origins of hypervelocity stars and to improve models of the Galactic potential.