  To elucidate the origin of re-enhanced high-$T_c$ phase in the heavily
electron-doped Fe-pnictides, systematic $^{75}$As NMR studies are performed on
heavily electron-doped LaFe$Pn$O$_{0.75}$H$_{0.25}$ by controlling the
pnictogen height ($h_{Pn}$) from the Fe plane through the substitution at
$Pn$(=As) site with Sb or P. The measurements of nuclear spin relaxation rate
(1/$T_1$) and Knight shift ($K$) reveal that the moderate spin fluctuations at
high temperatures are suppressed toward low temperatures. Such characteristic
spin fluctuations with gap like feature at low energies are more enlarged in
higher $T_c$ compounds with higher $h_{Pn}$, while those are totally suppressed
in non-superconducting compounds with lower $h_{Pn}$. This implies that the
contribution of the finite energy part in the spin fluctuation spectrum is
crucial for enhancing $T_c$ in the heavily electron-doped regime. This is in
contrast to many cases of typical Fe-based compounds with hole and electron
Fermi surfaces of similar sizes, where the spin fluctuations at low energies
develop significantly at low temperatures. The features in the heavily
electron-doped states are discussed in relation with the characteristics of the
faint hole Fermi surface derived from $d_{xy}$ orbital that rises when $h_{Pn}$
is high, together with the enhanced electron correlation effects.
