One group of people not to miss in examining secular morality are those presenting Moral Foundations Theory (MFT). See, in particular, Jonathan Haidt’s “The Righteous Mind” for a book length description of it. For a video presentation on the underlying approach to rationalism, see Haidt discussing “The Rationalist Delusion in Moral Psychology”.
There are three ideas to look for in MFT.
First, the existence of multiple, innate foundations of morality that generate conflict and stability. Both the innateness and multiple foundations in MFT undermine a rationalist approach to morality, although they do not eliminate the need for reasoning.
Second, rationalism is replaced with rationalization. We make snap decisions and then ask our brains to rationalize them.
Third, political tension is described as tension between different poles of these foundations. In particular, note how the tensions between individual rights and group loyalty are expressed across these foundations.
The theory is grounded in social psychology and neuroscience and it does not have a religious motivation although it is not hostile to religion. There is a politically liberal bias but MFT describes conservative moral positions as more balanced than those usually presented by liberals.