Hi,
This question has been on my mind for some time, but I struggle to find sources that detail with Jainism through a comparative approach or speak of its development (Maybe I'm not searching with the correct keywords.) Is Jainism a development from within Hinduism? Does its mythology share genetic (in the linguistic sense) affinity with Indo-European myths? Is it a "reform" movement? Or is Jainism a sort of pre-Indo-European tradition that was influenced and influenced hinduism during its development, especially in southern parts of India? The older sources I find do mention Jainism as rooted in pre-aryan traditions, but I can't find anything newer that talks about this. I'd also appreciate sources if it takes too long to write a response.
This is a great question! Because of that, the answer is complicated, and not all scholars agree on all of the details. But we can sketch the broad strokes of it. As you you seem to indicate in your question, at this time period (around 600 BCE), we can't really talk about 'Hinduism" as such, but rather Vedic religion broadly and Brahmanical orthodoxy and orthopraxy. Around this time period, 600 BCE (give or take a couple hundred years, depending on your dating), something called the "Shramana" movement begins to develop. Shramana means renouncer, and gets its name because these were religious/spiritual seekers who renounced household life, as well as renounced the primacy (or any influence at all) of the Vedas.
Whereas the Vedas and the dominant Brahmanical orthodoxy at the time were generally concerned with things that all ancient religions are concerned with-- fertility, bountiful harvests, successful progeny, etc.-- this Shramana movement starts to flip this on its head, and develops a whole theory and practice of asceticism. It is at this point that major elements of what we think of as "Hinduism" today develop and take shape, most notably the theory of rebirth. The focus of the Shramana movement is not how to be the most successful and prosperous in this life, but rather how to escape the chain of rebirth altogether. In very broad strokes, we could say that it became more interested in the transcendent than the mundane.
The roots of Jainism and Buddhism are located in this Shramana movement, as well as the Samkhya/Yoga philosophy that would be adopted back into Brahmanical orthodoxy. As to whether or not the Shramana movement is pre-Aryan, that is an even harder to answer. There were likely elements of pre-Aryan Dravidian religion at play in the Shramana movement, as well as Vedic and Brahmanical elements, as much as the Shramana movements tried to distance themselves from it. The story of the kind of simplistic Aryan invasion theory we also often hear about is itself also a lot more complicated.
In any event, I hope this was somewhat helpful. A really great resource to get started with might be Gavin Flood's Introduction to Hinduism, published by Cambridge University Press. He has a really good section on the Shramana movement, and its relationship to "Hinduism."