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In the present work, we propose a novel hybrid explicit jump immersed interface approach in conjunction with a higher order compact (HOC) scheme for simulating transient complex flows governed by the streamfunction-vorticity ($ψ$-$ζ$) formulation of the Navier-Stokes (N-S) equations for incompressible viscous flows. A new strategy has been adopted for the jump conditions at the irregular points across the interface using Lagrangian interpolation on a Cartesian grid. This approach, which starts with the discretization of parabolic equations with discontinuities in the solutions, source terms and the coefficients across the interface, can easily be accommodated into simulating flow past bluff bodies immersed in the flow. The superiority of the approach is reflected by the reduced magnitude and faster decay of the errors in comparison to other existing methods. It is seen to handle several fluid flow problems having practical implications in the real world very efficiently, which involves flows involving multiple and moving bodies. This includes the flow past a stationary circular and a twenty-four edge cactus cylinder, flows past two tandem cylinders, where in one situation both are fixed and in another, one of them is oscillating transversely with variable amplitude in time. To the best of our knowledge, the last two examples have been tackled for the first time by such an approach employing the $ψ$-$ζ$ formulation in finite difference set-up. The extreme closeness of our computed solutions with the existing numerical and experimental results exemplifies the accuracy and the robustness of the proposed approach.