Milne-like spacetimes are a class of k = -1 FLRW spacetimes which admit continuous spacetime extensions through the big bang. Moreover, the cosmological constant appears as an initial condition for Milne-like spacetimes under suitable assumptions on the scale factor. In this talk, we generalize this result to spacetimes which share similar geometrical properties with Milne-like spacetimes but without the strong isotropy assumption associated with them.