Karl Jaspers came very close to the eonic effect, but he couldn't quite find a 'second axial age' in the rise of the modern. Perhaps he was thrown off track by Nietzschean nihilism, or a postmodern disillusion with the First World War. Nietzsche is a misleading anti-modern reactionary, and quite confused about history. 

But the question of sacred and secular is irrelevant as we have seen. The modern transition is already almost strange for us and includes the period from Luther and the Reformation, the earliest stage of the Scientific Revolution in the time of Copernicus, and stretches to the Enlightenment, and its Divide period. This period is packed with rich material, and that includes the phenomenon of Protestantism. To call this nihilist is bad history, although the modern phenomenon of atheism is a genuine innovation, but not much more than 'still another dialectical product'. The real questions appear in their abstract form in Kant's Critique of Pure Reason, in his dialectic, where we see the mechanics of theism/atheism reduced to a series of 'ideas of Reason'. Religions of the future will involve a combination of 'pure practical reason' and the 'development of man's self-consciousness', which is about what they always were. This is perfectly compatible with science and secularism. 

There are a number of New Age formulations of the idea of a second Axial Age. But these are misleading and not based on sound historical thinking. No such New Age will occur, because the only candidate is the rise of the modern. A New Age Begins

There is nothing 'anti-spiritual' in this, in so far as such religions as Buddhism and Hinduism, influenced by the modernist surge of information, undergo strong revivals and diffusion. The rise of the modern was the best thing that happened to these religions, fortunately or unfortunately. 

However, the Enlightenment phenomenon provokes a strong critique of religious traditionalism. Might we point out that the Hebrew Prophets did the same for their era! Attempts to undermine this in a postmodern religious reaction can only create chaos. Note, for example, just how far Christianity and Judaism have traveled from their Axial sources. They are hardly even traditions, but modern deviations. Thus the real momentum lies in the roots of modernity itself. 

Note that the great religions spawned by the Axial era, such as Christianity, Islam, and Mahayana, all had their roots in the Axial interval. Any second Axial Age will thus work from sources in the modern transition. And that is completely natural. Note how the figure Hegel works to 'sublimate' Protestantism into modernity, and then Marx.... You might not agree with the result, but the gesture is profound and shows that the way the 'New Age' religious effect is at work in disguise.  

Thus the expectation of a second Axial Age that revives Axial religion is a prime fallacy. Our system will move on to something new. That future potential is completely obvious if we examine German Classical philosophy from Kant to Hegel/Schopenhauer, where the essence of Christianity and Buddhism/Upanishadism, is reworked on the spot. Nothing is preventing Buddhists or Hindus from doing what they always did. Modernity is not a totalitarian 'ism', but a pluralistic 'zoo' of many contradictions, mediating by the rationality of the Enlightenment. 

Thus we see our eonic phenomenon has already outsmarted the postmodern religious reactionaries.