You didn't mention if you're doing this digitization with a camera or a scanner.
I do it with a camera mainly, and one of the things I've noticed that orange-masked film is particularly affected by is stray light from not masking your backlight source perfectly. With slide film and a white light source, stray light is going to have the same color temperature as the light going through your film, and the more contrasty brightness range will mitigate it further. But with orange mask C-41, any stray light which is not being filtered through the orange mask is going to result in a different color temperature flare which may be near-impossible to mitigate in post processing. C-41 also has less contrast within each channel which means the effect of a flare will be greater since you are magnifying the contrast.
Assuming stray light is not the issue (you've masked and hooded things well, or are using a scanner), then if you are digitizing C-41 in 16-bits-per-channel it's important to look at the individual R-G-B histogram and make sure none of the channels are getting truncated. That will give you a RAW or TIF that you can work well with.
Having surmounted those two C-41 obstacles, you can invert, and then set your black point to the minimum of each of the R-G-B channels individually, and your white point to the maximum of the channels, in order to normalize the image and stretch the values over the full range available.
Then, you can pull the R, G, B curves up or down individually until you stop seeing unnatural color casts (unless desired for the effect). This is the part that takes a while if done manually, as I do in Gimp. Sometimes 5-10 minutes to get it right in an image I really care about. It does take practice to get the feel for what colors look natural as the eye adjusts. Usually pulling from the middle of the curves with a single point is sufficient, except in complicated cases. When you work on a whole roll of a particular type of film, you tend to see that you're making similar color adjustments on every image, unless some are badly under or over exposed. So it gets faster as you practice.
Of course, you can get more advanced than me and use software specifically designed to handle color negative, which may streamline some of these steps in some cases, at the loss of some manual control.