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We study the phase sensitivity of a Mach–Zehnder interferometer (MZI) with product detection (PD) scheme by using the superposition of Schrödinger’s cat-like state with the vacuum state (SCVS) and the coherent state as the inputs. We compare the results derived from PD with the parity detection scheme’s results. We found that PD performs better than parity detection and approaches the single parameter quantum Cramér-Rao bound in some cases. Therefore, we expect that SCVS may play as an alternative quantum resource with PD for the improvement in the phase sensitivity of an MZI having potential application in quantum metrology and quantum sensing.