Sorry I didn't get to this comment earlier!

The broadest answers to that question are Humanism and Scientific Pantheism, which I'm partial to. That's what a lot of atheists have made a whole lot of fun of.

Of course humanism is still sort of opposed to what I'm talking about in thr second paragraph,, or at least most humanists would be, as far as I'm aware. And most "earth religions" fall under some form of what I'd call spirituality.

In the U.S., at least, I've started attending my very local Unitarian Universalist church and I think they're your best actual practical bet. As much as I whine about most religion, I deeply respect the UU's commitment to include everyone, atheists included, and so I do my best to respect what people believe and what they want to talk about (which naturally doesn't usually involve trying to convert people). Third spaces are too important to keep people out because of some sense of sectarianism.

And, of course, Buddhism is at it's heart a very skeptical religion to the point that some interpret more as a philosophy, which is how I choose to see it, and it's a philosophy and a practice that modern psychology owes a whole lot to (and should probably yank even more from). It's literally just a framework of how to stop "suffering" and live a good life regardless of whether it seems like a good life externally. I do mostly stick to more secularized, almost new-age interpretations of it, I love the blog Deconstructing Yourself for being thoroughly dedicated to "Nondualism" while rejecting the schizo craziness it usually brings with it. But I like to learn from something closer to primary sources, too...

For which I'm relying on the Buddhist University. Of the two "original" explicitly supernatural elements of Buddhism, reincarnation was an assumption of the culture the Buddha was born into, and with that stripped away, Karma as cause and effect is just determinism without any "you're screwed because your past self screwed up". The second chapter of What the Buddha Taught (which is a great book) practically made me a Buddhist by illuminating just how dedicated the Buddha was to making sure people actually remained skeptical of him, only searching for what they could personally prove not really caring whether something was his idea or anyone else's. This is a quote (supposedly from him, not that it matters) that sums it up:

Yes, Kālāmas, it is proper that you have doubt, that you have perplexity, for a doubt has arisen in a matter which is doubtful. Now, look you Kālāmas, do not be led by reports, or tradition, or hearsay. Be not led by the authority of religious texts, nor by mere logic or inference, nor by considering appearances, nor the delight in speculative opinions, nor by seeming possibilities, nor by the idea: "this is our teacher". But, O Kālāmas, when you know for yourselves that certain things are unwholesome, and wrong, and bad, then give them up... And when you know for yourselves that certain things are wholesome and good, then accept them and follow them.

There's also a story about him telling off one of his followers for insisting that he reveal the "mysteries of the universe". He was pretty much like look, you're being a dumbass, that's won't help you live well, that's not important." I can respect that.