Is moral relativism the only alternative to God-inspired moral objectivism?

Please, let’s not turn this into a theism vs. atheism debate because that is truly not what I am interested in. But this something I was thinking about recently. Often the claim is made that without God-inspired moral objectivism, the only other alternative is complete moral relativism where moral actions are contingent upon culture and societies and even individuals.

But is that really the only other alternative? What other options are there for the interplay between atheism and morality? Is morality a religious construct? If it doesn’t exist objectively, then where does it exist? Is there a scientific neurological explanation for morality?