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We use a combination of numerical matrix product states (MPS) and analyticalapproaches to investigate the phase diagram of the two-leg $t$-$J$ ladder inthe region of low to intermediate fillings. We choose the same couplingstrength along the leg- and rung-directions, but study the effect of adding anearest-neighbor repulsion $V$. We observe a rich phase diagram andanalytically identify a Lifshitz-like band filling transition, which can beassociated to a numerically observed crossover from s-wave to d-wave likesuperconducting quasi-long range order (QLRO). Due to the strong interactions,the Lifshitz transition is smeared into a crossover region which separates twodistinct Luttinger theories with unequal physical meaning of the Luttingerparameter. Our numerically exact MPS results spotlight deviations from standardLuttinger theory in this crossover region and is consistent with Luttingertheory sufficiently far away from the Lifshitz transition. At very lowfillings, studying the Friedel-like oscillations of the local densityidentifies a precursor region to a Wigner crystal at small values of themagnetic exchange interaction $J/t$. We discuss analytically how tuningparameters at these fillings modifies the phase diagram, and find goodagreement with MPS results.

Lifshitz transition in the phase diagram of two-leg $t$-$J$ ladder systems at low filling