The International Celestial Reference Frame suffers from significantly less observations in the southern hemisphere compared to the northern one. One reason for this is the historically low number of very long baseline interferometry radio telescopes in the south. The AuScope very long baseline interferometry array with three new telescopes on the Australian continent and an identical antenna in New Zealand were built to address this issue. While the overall number of observations in the south has greatly improved since then, a closer look reveals that this improvement is only true for strong radio sources (source flux densities >0.6 Jy). The new array of small very long baseline interferometry antennas has a relatively low baseline sensitivity so that only strong sources can be observed within a short integration time. A new observing strategy, the star scheduling mode, was developed to enable efficient observations of weak sources during geodetic sessions, through the addition of a single more sensitive antenna to the network. This scheduling mode was implemented in the Vienna very long baseline interferometry Software and applied in four 24-h sessions in 2016. These observations provide updated positions and source flux densities for 42 weak southern radio sources and significantly reduce the formal uncertainties for these sources. The star scheduling mode now allows the AuScope very long baseline interferometry array to undertake greater responsibility in monitoring sources in the southern sky, without significantly weakening the session for geodetic purposes.