I make a screenshot of a part of the illustration and bring that to Photoshop (If you're working in Photoshop already you can select and copy all layers by hitting command-shift-C and pasting it in a new layer). Here I open the Hue/Saturation option from the Image > Adjustments menu and I will turn the Hue slider up and down. Often, something exciting happens and colours pop up that I wouldn’t have picked in Illustrator. In this case, I don't like what happens when I slide up, but I do like what happens when I bring the Hue slider down. When I am happy with a certain color, I take another screenshot and bring it back to illustrator, where I select the shapes with the same colour using the magic wand tool, replacing it with the new colour with the eyedropper tool. Usually it doesn’t work using all new colours. I often pick only one or two colours from the new screenshot, and then tweak my old colours based on the new colours, keeping the contrast up and the overall image fresh and appealing.
Looking for new and unique colours this way makes me think outside my comfort zone or signature palettes, and this process feels more natural than endlessly sliding CMYK or RGB sliders up and down.
Hope you get something out of this, regardless your style or software. Let me know how you handle colour in your work!