In $\mu$-hybrid inflation a nonzero inflaton vacuum expectation value induced by supersymmetry breaking is proportional to the gravitino mass $m_{3/2}$, which can be exploited to resolve the minimal supersymmetric standard model $\mu$ problem. We show how this scenario can be successfully implemented with $m_{3/2} \sim 1-100$~TeV and reheat temperature as low as $10^6$~GeV by employing a minimal renormalizable superpotential coupled with a well defined non-minimal K\"ahler potential. The tensor-to-scalar ratio $r$, a canonical measure of primordial gravity waves in most cases is less than or of the order of $10^{-6}-10^{-3}$.