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.