  We study the phenomenological consequences of recent results from atmospheric
and accelerator neutrino experiments, favoring normal neutrino mass ordering
$m_1 < m_2 < m_3$, a near maximal lepton Dirac CP phase $\delta_{l}\sim
270^\circ$ along with $\theta_{23}\gtrsim 45^\circ$, for possible realization
of natural structure in the lepton mass matrices characterized by ${({M_{ij}})}
\sim O(\sqrt {{m_i}{m_j}}) $ for $i,j=1,2,3$. It is observed that deviations
from parallel texture structures for $M_{l}$ and $M_{\nu}$ are essential for
realizing such structures. In particular, such hierarchical neutrino mass
matrices are not supportive for a vanishing neutrino mass $m_{\nu 1}\rightarrow
0$ characterized by Det$M_{\nu}\ne 0$ and predict ${m_{\nu 1}} \simeq (0.1 -
8.0)$ $meV$ , ${m_{\nu 2}} \simeq (8.0 - 13.0)$ $meV$, ${m_{\nu 3}} \simeq
(47.0 - 52.0)$ $meV$, $\Sigma \simeq (56.0 - 71.0)$ $meV$ and $\left\langle
{{m_{ee}}} \right\rangle \simeq (0.01 - 10.0)$ $meV$, respectively, indicating
that the task of observing a $0\nu \beta \beta$ decay may be rather challenging
for near future experiments.
