In addition to setting both the color temperature (blue←→amber axis) and tint (green←→magenta axis) I also often use an HSL (Hue-Saturation-Luminance) tool to remove or minimize remaining color casts. HSL tools divide the color spectrum into about eight different sectors and allow you to adjust the hue, saturation, and luminance of each sector independently of the others. Some applications call the tool HSV (Hue-Saturation-Value) or HSB (Hue-Saturation-Brightness) instead of HSL.

Adjustments made to white balance (color temperature and tint) pull pretty much all colors in the image in one direction or the other. Adjustments made in an HSL tool only affect the colors within the range of that single sector.

HSL tools can be used to remove color casts. They can also be used to create color casts or otherwise irregular color response curves. For instance, the examples linked in this question were shot under full spectrum daylight and then an HSL tool was used to alternately boost half the colors and pull back the other half as one travels around the color wheel. (Direct link here)

When shooting under any kind of flickering lighting, including fluorescent, each frame can have different brightness and color as the lights get brighter and dimmer due to the alternating current powering them. They tend to be bluer and fuller spectrum when at the brightest peak and browner and much more limited spectrum when at the dimmest part of the cycle. If you are using a shutter time shorter than half of the frequency of the current powering the lights, the color and brightness will change from the top to the bottom of the frame as the slit between the curtains of your focal plane shutter transits across the imaging sensor. Even with an electronic shutter you will see the effect with CMOS sensors, which scan across a sensor sequentially.

There are a few DSLRs now on the market that use the light meter to detect the timing of flickering lights and time the shutter to open when the lights are peaking.¹ This allows the photo to capture the image as the light is at both its brightest and fullest spectrum. Since the shutter opens at the brightest point in the cycle, it allows shorter shutter times for the same ISO and aperture settings. This can be quite an advantage when shooting sports under flickering lights. It also allows more uniformity from one frame to the next which simplifies the post processing workload.

Fluorescent lights are far from the only type of artificial lighting to be less than full spectrum and flicker as they vary due to the alternating current running through them. Here's an image taken under flickering stadium lights at night. The first version is using Auto White Balance. The second version is with the following adjustments made to white balance. The third is with the same WB adjustments and additional HSL adjustments.

The differences are subtle, to be sure, but a wide variety of skin tones and the color of the grass look more like what our eyes see after the HSL adjustments.

In this particular photo, the orange jacket was pretty much a lost cause when using only global adjustments. Skin tones are primarily in the orange band, with a lot of red as well. Yellow and magenta are also there to a lesser extent. Most of the color in grass, by the way, is usually almost all in the yellow, rather than green, band. To get the orange jacket looking natural using a global HSL tool would have totally wrecked the skin tones. Using a mask and adjusting that single area separately would be the only way to deal with it.

¹ Canon introduced their 'flicker reduction' feature in late 2014 with the 7D Mark II. The more recently introduced 5Ds, 5Ds R, 80D, 1D X Mark II, and 5D Mark IV also have it.