Spatially variable genes (SVGs) reveal the molecular and functional heterogeneity of cells across different spatial regions of a tissue. We found that sample-wide SVGs, identified by previous methods across the whole sample, largely overlap with cell-type marker genes derived from single-cell gene expression, leaving the spatial location information largely underutilized. We developed ctSVG, a computational method specifically tailored for Visium HD spatial transcriptomics at single-cell resolution. ctSVG accurately assigns Visium squares to cells and identifies cell-type-specific SVGs. We show that cell-type-specific SVGs identified by ctSVG include many new genes that do not overlap with sample-wide SVGs or cell-type marker genes, and that these genes reveal important biological functions in real spatial datasets.