  A weird electronic state accompanied with an anomalous superconducting
precursor and/or exotic orders, called the pseudogap state, arises prior to a
superconducting condensate in underdoped cuprates that are situated near Mott
transition. Another way to make the system approach the Mott transition is the
variation of bandwidth or correlation strength, which gives a new dimension to
exploring this exotic state. Here we report nuclear magnetic resonance (NMR)
studies on layered organic superconductors with half-filled bands whose widths
are pressure-tuned near the Mott transition. The system situated on the verge
of the Mott transition shows a pseudogap-like anomalous suppression of spin
excitations on cooling from well above the superconducting critical temperature
$T_{\mathrm{c}}$. The pressure variation of the NMR relaxation rate shows that
the pseudogap-like behavior is rapidly suppressed by applying pressure. The NMR
experiments under various magnetic fields varied up to 18 T proves the absence
of symmetry breaking orders that compete with superconductivity, such as charge
orders, in the metallic phase. Remarkably, the pseudogap-like behavior above
$T_{\mathrm{c}}$ and the superconducting condensate fade out in parallel under
ascending magnetic fields with similar field-orientation dependence, indicating
a superconducting precursor is the predominant origin of the pseudogap. Our
further investigation of different materials, which take different "distances"
from the Mott transition by chemical pressure, confirms that the
superconducting precursor is not the conventional amplitude fluctuations
arising from low dimensionality but unconventional preformation of Cooper pairs
enhanced near the Mott transition. These findings conclude that preformed
Cooper pairs persist up to twice as high as $T_{\mathrm{c}}$ on the verge of
the bandwidth-controlled Mott transition.
