  We consider the nuclear effective field theory including pions in the
two-nucleon sector in the S waves up to including the next-to-next-to-leading
order (NNLO) terms according to the power counting suggested by the Wilsonian
renormalization group analysis done in a previous paper. We treat only the
leading contact interaction nonperturbatively, and the rest, including the
long-distance part of pion exchange, are treated as perturbations. To define
the long-distance part, it is important to introduce a separation scale, or a
cutoff. We employ a hybrid regularization, in which the loops with only contact
interactions are regularized with Power Divergence Subtraction (PDS), while the
loops with (long-distance part of) pion exchange are regularized with a
Gaussian damping factor (GDF), to simplify the (nonperturbative) leading-order
amplitudes. The scale introduced by PDS is identified with the cutoff of GDF up
to a numerical factor. We emphasize that the introduction of the GDF requires a
careful definition of the coupling constant for the pion exchange. We obtain
the analytic expressions for the phase shifts for the $^1S_0$ and
$^3S_1$-$^3D_1$ channels. By fitting them to the Nijmegen partial wave analysis
data, it is shown that the effective theory expansion with perturbative
long-distance part of pion exchange is converging.
