Streptococcus pyogenes Cas9 (SpCas9) has been employed as a genome engineering tool with a promising potential within therapeutics. However, its off-target effects present major safety concerns for applications requiring high specificity. Approaches developed to date to mitigate this effect, including any of the increased-fidelity (i.e., high-fidelity) SpCas9 variants, only provide efficient editing on a relatively small fraction of targets without detectable off-targets. Upon addressing this problem, we reveal a rather unexpected cleavability ranking of target sequences, and a cleavage rule that governs the on-target and off-target cleavage of increased-fidelity SpCas9 variants but not that of SpCas9-NG or xCas9. According to this rule, for each target, an optimal variant with matching fidelity must be identified for efficient cleavage without detectable off-target effects. Based on this insight, we develop here an extended set of variants, the CRISPRecise set, with increased fidelity spanning across a wide range, with differences in fidelity small enough to comprise an optimal variant for each target, regardless of its cleavability ranking. We demonstrate efficient editing with maximum specificity even on those targets that have not been possible in previous studies.