Aims: We present new measurements of the abundances of carbon and oxygen derived from high-excitation C i and O i absorption lines in metal-poor halo stars, with the aim of clarifying the main sources of these two elements in the early stages of the chemical enrichment of the Galaxy. Methods: We target 15 new stars compared to our previous study, with an emphasis on additional C/O determinations in the crucial metallicity range -3 ⪉ [Fe/H]⪉ -2. The stellar effective temperatures were estimated from the profile of the Hβ line. Departures from local thermodynamic equilibrium were accounted for in the line formation for both carbon and oxygen. The non-LTE effects are very strong at the lowest metallicities but, contrary to what has sometimes been assumed in the past due to a simplified assessment, of different degrees for the two elements. In addition, for the 28 stars with [Fe/H] < -1 previously analysed, stellar parameters were re-derived and non-LTE corrections applied in the same fashion as for the rest of our sample, giving consistent abundances for 43 halo stars in total. Results: The new observations and non-LTE calculations strengthen previous suggestions of an upturn in C/O towards lower metallicity (particularly for [O/H] ⪉ -2). The C/O values derived for these very metal-poor stars are, however, sensitive to excitation via the still poorly quantified inelastic H collisions. While these do not significantly affect the non-LTE results for C i, they greatly modify the O i outcome. Adopting the H collisional cross-sections estimated from the classical Drawin formula leads to [C/O] ≈ 0 at [O/H] ≈ -3. To remove the upturn in C/O, near-LTE formation for O i lines would be required, which could only happen if the H collisional efficiency with the Drawin recipe is underestimated by factors of up to several tens of times, a possibility which we consider unlikely. Conclusions: The high C/O values derived at the lowest metallicities may be revealing the fingerprints of Population III stars or may signal rotationally-aided nucleosynthesis in more normal Population II stars. Based on data collected with the European Southern Observatory's Very Large Telescope (VLT) at the Paranal, Chile (programmes No. 67.D-0106 and 73.D-0024) and with the Magellan Telescope at Las Campanas Observatory, Chile.