In this work, we explore spherically symmetric black-hole solutions in a specific subclass of the Horndeski theories. By adopting a potential engineering approach, we derive exact analytic black-hole solutions endowed with scalar hair. We analyze the properties of these novel solutions and discover a spectrum that encompasses normal, ultra-compact, and ultra-sparse black holes. Each type of black hole emerges from a distinct range of scalar charge values. Rotating versions of these black holes are also obtained through a perturbation method, and their relative angular velocities are calculated. These rotation signatures may offer potential observational probes to test gravity. Our solutions showcase the richness of black hole geometries permitted in scalar-tensor theories beyond General Relativity. Further efforts to establish stability and astrophysical viability will be important in future directions.