  Second-order topological insulator (SOTI) is featured with the presence of
$(d-2)$-dimensional boundary states in $d$-dimension systems. The
non-Hermiticity induced breakdown of bulk-boundary correspondence (BBC) and the
periodic driving on systems generally obscure the description of non-Hermitian
SOTI. To prompt the applications of SOTIs, we explore the role of periodic
driving in controllably creating exotic non-Hermitian SOTIs both for 2D and 3D
systems. A scheme to retrieve the BBC and a complete description to SOTIs via
the bulk topology of such nonequilibrium systems are proposed. It is found that
rich exotic non-Hermitian SOTIs with a widely tunable number of 2D corner
states and 3D hinge states and a coexistence of the first- and second-order
topological insulators are induced by the periodic driving. Enriching the
family of topological phases, our result may inspire the exploration to apply
SOTIs via tuning the number of corner/hinge states by the periodic driving.
