While molecular quasar absorption systems provide unique probes of the physical and chemical properties of the gas as well as original constraints on fundamental physics and cosmology, their detection remains challenging. Here we present the results from a complete survey for molecular gas in thirty-nine absorption systems selected solely upon the detection of neutral carbon lines in Sloan Digital Sky Survey (SDSS) spectra, without any prior knowledge of the atomic or molecular gas content. H₂ is found in all twelve systems (including seven new detections) where the corresponding lines are covered by the instrument setups and measured to have logN(H₂) ≳ 18, indicating a self-shielded regime. We also report seven CO detections (7/39) down to logN(CO) ~ 13.5, including a new one, and put stringent constraints on N(CO) for the remaining 32 systems. N(CO) and N(C I) are found to be strongly correlated with N(CO)/N(C I) ~ 1/10. This suggests that the C I-selected absorber population is probing gas deeper than the H I–H₂ transition in which a substantial fraction of the total hydrogen in the cloud is in the form of H₂. We conclude that targeting C I-bearing absorbers is a very efficient way to find high-metallicity molecular absorbers. However, probing the molecular content in lower-metallicity regimes as well as high-column-density neutral gas remains to be undertaken to unravel the processes of gas conversion in normal high-z galaxies.