  After extensive investigation on the Floquet second-order topological
insulator (FSOTI) in two dimension (2D), here we propose two driving schemes to
systematically engineer the hierarchy of Floquet first-order topological
insulator, FSOTI, and Floquet third-order topological insulator in three
dimension (3D). Our driving protocols allow these Floquet phases to showcase
regular $0$, anomalous $\pi$, and hybrid $0$-$\pi$-modes in a unified phase
diagram, obtained for both 2D and 3D systems, while staring from the lower
order topological or non-topological phases. Both the step drive and the mass
kick protocols exhibit the analogous structure of the evolution operator around
the high symmetry points. These eventually enable us to understand the Floquet
phase diagrams analytically and the Floquet higher order modes numerically
based on finite size systems. The number of $0$ and $\pi$-modes can be tuned
irrespective of the frequency in the step drive scheme while we observe
frequency driven topological phase transitions for the mass kick protocol. We
topologically characterize some of these higher order Floquet phases (harboring
either $0$ or anomalous $\pi$ mode) by suitable topological invariant in 2D and
3D cases.
