Context. In addition to the systematic observations of known solar-system objects (SSOs), a continuous processing of new discoveries requiring fast responses is implemented as the short-term processing of Gaia SSO observations, providing alerts for ground-based follow-up observers. The common independent observation approach for the purposes of orbit computation has led to unrealistically large ephemeris prediction uncertainties when processing real Gaia data. Aims. We aim to provide ground-based observers with a cloud of sky positions that is shrunk to a fraction of the previously expected search area by making use of the characteristic features of Gaia astrometry. This enhances the efficiency of Gaia SSO follow-up network and leads to an increased rate of asteroid discoveries with reasonably constrained orbits with the help of ground-based follow-up observations of Gaia asteroids. Methods. We took advantage of the separation of positional errors of Gaia SSO observations into a random and systematic component. We treated the Gaia observations in an alternative way by collapsing up to ten observations that correspond to a single transit into a single so-called normal point. We implemented this input procedure in the Gaia SSO short-term processing pipeline and the OpenOrb software. Results. We validate our approach by performing extensive comparisons between the independent observation and normal point input methods and compare them to the observed positions of previously known asteroids. The new approach reduces the ephemeris uncertainty by a factor of between three and ten compared to the situation where each point is treated as a separate observation. Conclusions. Our new data treatment improves the sky prediction for the Gaia SSO observations by removing low-weight orbital solutions. These solutions originate from excessive curvature of observations, introduced by short-term variations of Gaia attitude on the one hand, and, as a main effect, shrinking of systematic error bars in the independent observation case on the other hand. We anticipate that a similar approach may also be utilized in a situation where observations from a single observatory dominate.