There is never the need to tear off the coil and start over unless the coil has burned out or it's really old. Your best bet is to always work with the one that's on it, because the coil becomes easier and easier to manipulate as it ages in... never takes more than a few days usually, and once you get good at it, it's much quicker.
Also, a common misconception is that you need to get every coil glowing... well.. when the wick is brand new with no
juice on it, that can happen... but if there is juice in the tank, the bottom coil or two may never glow because there's juice on it.
And Jen, it's not just you... people new at this need to resist the urge to start over when it's acting funny. I wonder if this is in the FAQ... it certainly should be!