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This paper deals with constrained trace,matrix and constrained matrix Hack inequalities for the nonlinear heat equation ωt =△ω + aω ln ω on closed manifolds.A new interpolated Hack inequality for ωt =△ω-ω ln ω+eRω on closed surfaces under ε-Ricci flow is also derived.Finally,the author proves a new differential Hack inequality for ωt =△ω-ω ln ω under Ricci flow without any curvature condition.Among these Hack inequalities,the correction terms are all time-exponential functions,which are superior to time-polynomial functions.