We propose a novel reformulation of supersymmetric (more precisely μ-) hybrid inflation based on a local U(1) or any suitable extension of the minimal supersymmetric standard model (MSSM) which also resolves the μ problem. We employ a suitable Kahler potential which effectively yields quartic inflation with non-minimal coupling to gravity. Imposing the gravitino Big Bang Nucleosynthesis (BBN) constraint on the reheat temperature (({T}_{r}\lesssim {10}^{6}) GeV) and requiring a neutralino LSP, the tensor to scalar ratio (r) has a lower bound (r\gtrsim 0.004). The U(1) symmetry breaking scale lies between 10(^{8}) and 10(^{12}) GeV. We also discuss a scenario with gravitino dark matter whose mass is a few GeV.