Milne-like spacetimes are a class of k = -1 FLRW spacetimes which admit continuous extensions through the big bang. Under suitable assumptions on the scale factor, it was previously shown that the cosmological constant appears as an initial condition for Milne-like spacetimes; this yields a "quasi de Sitter" expansion for the early universe which could have applications to inflationary scenarios. In this talk, we generalize these results to spacetimes which share similar geometrical properties with Milne-like spacetimes but without any spatially isotropic or homogeneous assumptions.