I usually just rinse/ wet my old wick under the faucet, that usually makes it slide out. At least with my coil and rayon set ups, if the rinse doesn't make my old wick come out, it usually means I put a little more then I should have. Now cotton is slightly different, in that it kind of swells compared to the rayon.
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See, things like this i kind of overlook, but is a perfectly good solution. I had got into the habit of not getting my coils anywhere near water for fear of having them go rusty. Only recently i have been thinking that as long as i dry them out with a burn or two then it shouldn't matter at all.
I'm going to carefully rinse it in future.