From my experience, even for dark blue, you'll pretty much have to bleach at least the roots if you want it to be vibrant. And for blues, you really want to get it quite light, otherwise you risk having it turn out muddy or even green. You *might* be able to get away with leaving the part that's already purple alone, though, especially if you let it fade out a bit more first. I did exactly that once or twice, and couldn't see much or any difference between the newly bleached roots and the purple I was covering (same thing with a brighter magenta, even).
I second the suggestion to use a cheap "clarifying" shampoo like Suave to help strip some of the dye out, though. There are other methods that are less damaging than bleach, but still more damaging than a harsh shampoo, so try the shampoo first. After that you can test the dye on a little section of roots and a little section of the purple to see how it turns out before doing anything more extreme.
If you don't want to go as crazy with the bleach, you can pick a warmer colour, as long as it's not super pale. So, stick with something like purple, or switch to magenta, red, orange... Those are usually more forgiving if you don't strip out all of your natural colour to the point that it's platinum, because what's going to be left after lightening will be a warm tone to begin with.
Also from experience, whatever you do, don't do anything you see here, unless you're also fine with shaving your head:
https://www.youtube.com/channel/UCA45gb-CufIUq3Vv0AqkeKQ
