Initial google searches are stating that you obtain more spirit iron chunks in nioh by disassembling divine items, but i'm only ever getting divine fragments from it. I had several hundred at one point, which i've blown through, and haven't figured out how to obtain more.
I believe you get spirit iron chunks from disassembling rare items (blue) and spirit iron fragments from uncommon items (yellow). You also get two each time for completing the very short side mission "The Battle of Ohashi" in the Kinki region. It's completed by defeating only one enemy, so that's probably the fastest way.