  Much having explored on two-dimensional higher-order topological
superconductors (HOTSCs) hosting Majorana corner modes (MCMs) only, we propose
a simple fermionic model based on a three-dimensional topological insulator
proximized with $s$-wave superconductor to realize Majorana hinge modes (MHMs)
followed by MCMs under the application of appropriate Wilson-Dirac
perturbations. We interestingly find that the second-order topological
superconductor, hosting MHMs, appear above a threshold value of the first type
perturbation while third-order topological superconducting phase, supporting
MCMs, immediately arises incorporating infinitesimal perturbation of the second
kind. Thus, a hierarchy of HOTSC phases can be realized in a single
three-dimensional model. Additionally, the application of bulk magnetic field
is found to be instrumental in manipulating the number of MHMs, leaving the
number for MCMs unaltered. We analytically understand these above mentioned
numerical findings by making resort to the low energy model. We further
topologically characterize these phases with a distinct structure of the
Wannier spectra. From the practical point of view, we manifest quantized
transport signatures of these higher-order modes. Finally, we construct Floquet
engineering to generate the ladder of HOTSC phases by kicking the same
perturbations as considered in their static counterpart.
