Classical Solar System tests of photons coupled to Weyl tensor with two polarizations were studied in a recent work. A coupling strength parameter a in this model was firstly obtained as |alpha| less than or similar to 4 x 10(11) m(2) by using available datasets in the Solar System. In this paper, a new test called by gravitational time advancement is proposed and investigated to test such the coupling. This new test, which is quite different from Shapiro time delay, depends strongly on round-trip proper time span (not coordinate time one) of flight of radio pulses between an observer on the Earth and a distant spacecraft. For ranging a spacecraft getting far away from the Sun, two special cases (the superior/inferior conjunctions) are used to analyse the observability in the advancement contributed by the Weyl coupling. We found that the situation of the inferior conjunction is more suitable for detecting the advancement caused by such the Weyl coupling. In either case, two kinds of polarizations make the advancement in the model smaller or larger than the one of general relativity. Although the observability in the advancement could be out of the reach of already existing technology, the implement of planetary laser ranging and optical clocks might provide us more insights on such the Weyl coupling in the near future.