We report on measurements of the magnetic susceptibility and electrical resistance under He-gas pressure on single crystals of Ca(Fe1−xCox)2As2. 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≤ Ts,N) and superconducting (T≤Tc) states against variation of pressure, disclosing pressure coefficients of dTs,N/dP=−(1100±50) K/GPa and dTc/dP=−(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.