The Chromospheric Lyman-Alpha Spectro-Polarimeter is a sounding rocket experiment that has provided the first successful measurement of the linear polarization produced by scattering processes in the hydrogen Lyα line (121.57 nm) radiation of the solar disk. In this paper, we report that the Si iii line at 120.65 nm also shows scattering polarization and we compare the scattering polarization signals observed in the Lyα and Si iii lines in order to search for observational signatures of the Hanle effect. We focus on four selected bright structures and investigate how the U/I spatial variations vary between the Lyα wing, the Lyα core, and the Si iii line as a function of the total unsigned photospheric magnetic flux estimated from Solar Dynamics Observatory/Helioseismic and Magnetic Imager observations. In an internetwork region, the Lyα core shows an antisymmetric spatial variation across the selected bright structure, but it does not show it in other more magnetized regions. In the Si iii line, the spatial variation of U/I deviates from the above-mentioned antisymmetric shape as the total unsigned photospheric magnetic flux increases. A plausible explanation of this difference is the operation of the Hanle effect. We argue that diagnostic techniques based on the scattering polarization observed simultaneously in two spectral lines with very different sensitivities to the Hanle effect, like Lyα and Si iii, are of great potential interest for exploring the magnetism of the upper solar chromosphere and transition region.