  X-ray assisted nuclear excitation by electron capture (NEEC) into inner-shell
atomic holes in a plasma environment generated by strong optical lasers is
investigated theoretically. The considered scenario involves the interaction of
a strong optical laser with a solid-state nuclear target leading to the
generation of a plasma. In addition, intense x-ray radiation from an X-ray Free
Electron Laser (XFEL) produces inner-shell holes in the plasma ions, into which
NEEC may occur. As case study we consider the $4.85$-keV transition starting
from the 2.4 MeV long-lived $^{\mathrm{93m}}$Mo isomer that can be used to
release the energy stored in this metastable nuclear state. We find that the
recombination into $2p_{1/2}$ inner-shell holes is most efficient in driving
the nuclear transition. Already at few hundred eV plasma temperature, the
generation of inner-shell holes can allow optimal conditions for NEEC,
otherwise reached for steady-state plasma conditions in thermodynamical
equilibrium only at few keV. The combination of x-ray and optical lasers
presents two advantages: first, NEEC rates can be maximized at plasma
temperatures where the photoexcitation rate remains low. Second, with mJ-class
optical lasers and an XFEL repetition rate of $10$ kHz, the NEEC excitation
number can reach $\sim 1$ depleted isomer per second and is competitive with
scenarios recently envisaged at petawatt-class lasers.
