Carbon-rich grains are observed to condense in the ejecta of recent core-collapse supernovae (SNe) within a year after the explosion. Silicon carbide grains of type X are C-rich grains with isotopic signatures of explosive SN nucleosynthesis have been found in primitive meteorites. Much rarer silicon carbide grains of type C are a special sub-group of SiC grains from SNe. They show peculiar abundance signatures for Si and S, isotopically heavy Si, and isotopically light S, which appear to be in disagreement with model predictions. We propose that C grains are formed mostly from C-rich stellar material exposed to lower SN shock temperatures than the more common type X grains. In this scenario, extreme S enrichments observed in C grains may be explained by the presence of short-lived Si (τ = 153 yr) in the ejecta, produced by neutron capture processes starting from the stable Si isotopes. No mixing from deeper Si-rich material and/or fractionation of Si from S due to molecular chemistry is needed to explain the S enrichments. The abundance of Si in the grains can provide constraints on the neutron density reached during the SN explosion in the C-rich He shell material. The impact of the large uncertainty of the neutron capture cross sections in the Si region is discussed.