  Milne-like spacetimes are a class of $k = -1$ FLRW spacetimes which admit
continuous spacetime extensions through the big bang. In a previous paper [30],
it was shown that the cosmological constant appears as an initial condition for
Milne-like spacetimes. In this paper, we generalize this statement to
spacetimes which share similar geometrical properties with Milne-like
spacetimes but without the strong spatially isotropic assumption associated
with them. We show how our results yield a "quasi de Sitter" expansion for the
early universe which could have applications to inflationary scenarios.
