Jazz in noir is hardly important for understanding its origins--it is a blip on the horizon until the mid-50s, grafted onto the "nightlife" culture that is often portrayed as part of (or at least adjacent to) the criminal milieu. Noir is about alienation first and foremost, with crime often the result (though not always, as the "melo-noirs" of the 40s and 50s demonstrate); this core structure never changes until we get to a flashpoint when politics become more foregrounded in the narrative of these films, with "neo-noir" correcting this either by a shorthand approach to cultural/political underpinnings or an "over-determination" made by attempting to appropriate the past ("retro-neo").
Once the "heist" model becomes prevalent in late noir, it pervades a good deal of the crime film landscape, creating a series of character/narrative interactions that become more overtly formulaic, leading to an increasing use of extremity in characters/action/sex/violence/soundtrack, which ultimately undercuts the shock value inherent in classic noir.
Fortunately, the noir films made in Europe and Japan (and South America) from the late 50s to the mid 60s capture their own cultural ethos while engaging with the alienation understructures that anchor the classic noir narrative. That final flowering of classic noir is what we should be focusing on, because it shows us where the "rift zone" of modernity lies, and how we've fallen into the post-modern abyss that the world is having an incredibly tough time escaping.