Time symmetric block time step algorithm (TSBTS) is a newly developed efficient method for N-body integration algorithms. It is aimed to collect time symmetric behavior of the integrations with block time step mentality. This algorithm is firstly developed with second order structurally time symmetric leapfrog integration scheme. In this work, we aimed to show that, this algorithm gives very well results with time symmetric fourth-order Hermite integration scheme. We also show that, the algorithm algorithm protect high energy conservation skill when order we use it with sixth and eighth order Hermite schemes without time symmetric interpolations.