  Inspired by the observation of a robust $d$-wave superconducting phase driven
by tuning the next-nearest-neighbor (NNN) electron hopping in recent density
matrix renormalization group (DMRG) studies of six- and eight-leg $t$-$J$
model, we systematically study the phase diagram of the two-leg $t$-$J$ ladder
with the NNN couplings (including NNN hopping and spin interaction) in a large
region of doping level, by means of the DMRG calculations. Upon doping from
half filling, we identify the Luther-Emery liquid (LEL) phase, which can be
distinguished as the pairing-dominant and charge density-dominant regime by
comparing the Luttinger parameter $K_{\rho}$. With the growing NNN couplings,
pairing correlations are enhanced and correspondingly $K_{\rho}$ increases,
driving the system from the density-dominant to the pairing-dominant regime. In
the Tomonaga-Luttinger liquid (TLL) phase in the larger doping region, we
identify two TLL regimes with different features of charge density correlation.
At the quarter filling ($1/2$ doping level), we find that the strong dimer
orders of bond energy in the open system actually decay algebraically and thus
do not indicate a spontaneous translational symmetry breaking. Our results show
that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the
similar role as that on the wider $t$-$J$ cylinder, and studies on this more
accessible system can be helpful towards understanding the emergence of the
remarkable $d$-wave superconducting phase on the wider system.
