Six candidate overlapping genes have been detected in SARS-CoV-2, yet current methods struggle to detect overlapping genes that recently originated. However, such genes might encode proteins beneficial to the virus, and provide a model system to understand gene birth. To complement existing detection methods, I first demonstrated that selection pressure to avoid stop codons in alternative reading frames is a driving force in the origin and retention of overlapping genes. I then built a detection method, CodScr, based on this selection pressure. Finally, I combined CodScr with methods that detect other properties of overlapping genes, such as a biased nucleotide and amino acid composition. I detected two novel ORFs (ORF-Sh and ORF-Mh), overlapping the spike and membrane genes respectively, which are under selection pressure and may be beneficial to SARS-CoV-2. ORF-Sh and ORF-Mh are present, as ORF uninterrupted by stop codons, in 100% and 95% of the SARS-CoV-2 genomes, respectively.