There is nothing in atheism that does promote better morality, or much of anything else. As I’ve said elsewhere, atheism is virtually content free.
Having said that, the objective basis for morality in a Godless Universe is the well-being of conscious creatures, as realized in states of the brain and events in the Universe. This is not arbitrary, for one thing. Literally everything we do, every choice we make, in this Universe, is bent on improving our conscious experience. Nor is it subjective in the sense that it’s “just my opinion.” It is virtually everyone’s opinion, religious and irreligious alike, whether they admit it or not. It is, granted, a self-justifying assumption, but that does nothing to invalidate it. The science of medicine begins with a self-justifying assumption - that the goal of the science is to improve physical health. The game of chess begins with the self-justifying assumption that the goal of a any competitive game is to checkmate your opponent. None of us are in anyway burdened by the bootless cries of “Says who?” when someone tries to attack the philosophical underpinnings of chess or medicine. If you claim to be playing chess in such a way that has nothing to do with checkmating your opponent, you’re not playing chess. If you claim your medical practice has nothing to do with improving anyone’s physical health, you are guilty of misusing the term “medicine.” And if you say that human morality has nothing to with questions of human well-being and suffering, I accuse you of not knowing what you’re talking about, or pretending not to.