...went wild and crazy and dumped the whole thing the dye pot.
Well, the original color was kind of dull, a grayish beige with faint tweedy bits of pale blue and lilac. It's a blend of silk and cotton, so I knew that both fibers would react to the dye differently, enhancing the tweedy look. After soaking in vinegar and water, I dyed it with a little bit of blue at a time, and drizzled some dilute purple onto the yarn in the bowl, making subtle purple variegation here and there.
The results: a pale gray-blue tweed with lilac highlights.
Now, with a new dye job, the yarn's karma should be sufficiently re-set to tackle this pattern again.