This paper investigates the design of a finite-time hybrid bumpless transfer fault detection filter(FDF) for switched linear systems with all subsystems unstable under mode-dependent interval dwell time(MDIDT) switching. Firstly, in order to improve the transient performance of the fault detection system, a hybrid bumpless transfer performance description is proposed, which directly limits the total amplitude of residual and rate bumps at switching instants. Then, by dividing the dwell time interval into segments, a less conservative time-varying discretized Lyapunov function is constructed, and sufficient conditions for ensuring the finite-time stability and finite-time $H_\infty$ performance of the augmented system under MDIDT switching are obtained. On this basis, the design conditions of the finite-time hybrid bumpless transfer FDF are given. Finally, a numerical example is given to verify the effectiveness and superiority of the proposed method.