  We use a combination of numerical matrix product states (MPS) and analytical
approaches to investigate the phase diagram of the two-leg $t$-$J$ ladder in
the region of low to intermediate fillings. We choose the same coupling
strength along the leg- and rung-directions, but study the effect of adding a
nearest-neighbor repulsion $V$. We observe a rich phase diagram and
analytically identify a Lifshitz-like band filling transition, which can be
associated to a numerically observed crossover from s-wave to d-wave like
superconducting quasi-long range order (QLRO). Due to the strong interactions,
the Lifshitz transition is smeared into a crossover region which separates two
distinct Luttinger theories with unequal physical meaning of the Luttinger
parameter. Our numerically exact MPS results spotlight deviations from standard
Luttinger theory in this crossover region and is consistent with Luttinger
theory sufficiently far away from the Lifshitz transition. At very low
fillings, studying the Friedel-like oscillations of the local density
identifies a precursor region to a Wigner crystal at small values of the
magnetic exchange interaction $J/t$. We discuss analytically how tuning
parameters at these fillings modifies the phase diagram, and find good
agreement with MPS results.
