It depends on your definition of moral. Religions define moral as keeping to things that will neither hurt you nor the society at large. Some of these things are merely encouraged while not doing some are criminalized.
For example, the multiplier effect of fornication and adultery is enormous. Religion believes that it is better to prevent it by criminalizing it rather than treating the effects.
In other words, you cant just say that secular societies are the ones pushing religion towards the right moral part without defining what moral is.
By moral I meant what most societies accept as moral, and mostly in relation to the past. Like, most people will now accept that being gay is fine, slavery is bad, killing people to convert their religion is bad, killing people cos you can see more than their eyes is bad, pedophilia is bad, etc. According to the books of the biggest religions I know, those things were perfectly fine, so I don't see how religion cares about the individual or society.
I don't think I need to define 'moral' in order to make a historical claim. I think humanity by and large progresses against religion, not because of religion. Religion is actually what's holding humanity back.
But, like I said, I'm an atheist, so of course I would believe these things! A proper discussion would start from 'does God exist?' and 'can we have objective morality without God?' and things like that. Mostly I just wanted to say that there are no examples of non-religious societies that have no guiding principles. It's impossible for any conscious beings to lack values. All societies have values. Secular societies are proven to be much better and happier than theocratic societies.