Despite the general agreement on the significance of T cells during SARS-CoV-2 infection, the clinical impact of specific and cross-reactive T-cell responses remains uncertain. Understanding this aspect could provide insights for adjusting vaccines and maintaining robust long-term protection against continuously emerging variants. To characterize CD8+ T-cell response to SARS-CoV-2 epitopes unique to the virus (SC2-unique) or shared with other coronaviruses (CoV-common), we trained a large number of T-cell receptor (TCR) - epitope recognition models for MHC-I-presented SARS-CoV-2 epitopes from publicly available data. These models were then applied to longitudinal CD8+ TCR repertoires from critical and non-critical COVID-19 patients. In spite of comparable initial CoV-common TCR repertoire depth and CD8+ T-cell depletion, the temporal dynamics of SC2-unique TCRs differed depending on the disease severity. Specifically, while non-critical patients demonstrated a large and diverse SC2-unique TCR repertoire by the second week of the disease, critical patients did not. Furthermore, only non-critical patients exhibited redundancy in the CD8+ T-cell response to both groups of epitopes, SC2-unique and CoV-common. These findings indicate a valuable contribution of the SC2-unique CD8+ TCR repertoires. Therefore, a combination of specific and cross-reactive CD8+ T-cell responses may offer a stronger clinical advantage. Besides tracking the specific and cross-reactive SARS-CoV-2 CD8+ T cells in any TCR repertoire, our analytical framework can be expanded to more epitopes and assist in the assessment and monitoring of CD8+ T-cell response to other infections.