We report on measurements of the magnetic susceptibility and electrical resistance under He-gas pressure on single crystals of Ca(Fe 1 − x Co x ) 2 As 2 . We find that for properly heat-treated crystals with modest Co concentration, x = 0.028, the salient ground states associated with iron-arsenide superconductors, i.e., orthorhombic/antiferromagnetic (o/afm), superconducting, and nonmagnetic collapsed-tetragonal (cT) states can be accessed all in one sample with reasonably small and truly hydrostatic pressure. This is possible owing to the extreme sensitivity of the o/afm (for T ≤ T s , N ) and superconducting ( T ≤ T c ) states against variation of pressure, disclosing pressure coefficients of d T s , N / d P = − ( 1100 ± 50 ) K/GPa and d T c / d P = − ( 60 ± 3 ) K/GPa, respectively. Systematic investigations of the various phase transitions and ground states via pressure tuning revealed no coexistence of bulk superconductivity (sc) with the o/afm state which we link to the strongly first-order character of the corresponding structural/magnetic transition in this compound. Our results, together with literature results, indicate that preserving fluctuations associated with the o/afm transition to low enough temperatures is vital for sc to form.