A partial solution is always better than no solution. If you want to change yourself, focus on congratulating yourself for partial success instead of punishing yourself for incomplete success; then the success will grow. Let me re-frame it this way: You already have a solution that works (wrapping your fingers), and that's great! Honestly; it's an improvement over the time where you didn't have even that solution. Many people are still there; but you are not. You just want even better solution.
Now the same approach on the micro level: If you happen to find yourself biting your finger (wrapped or not), should you be angry at yourself? No. You should congratulate yourself for noticing. Of course you should immediately stop doing it. And you should congratulate yourself for stopping.
The idea is that you get more of what you reward; which is why you want to reward noticing and stopping, because you want more noticing and stopping. -- Punishing yourself for doing something wrong seems to make sense... until you realize that you are effectively punishing yourself for noticing that you are doing something wrong (because when you didn't notice it, there was no punishment, was it?). So you definitely want to reward noticing.
(What is a "reward" in this context? Whatever works for you. Give yourself a mental high five; think "I am great" or "I am improving, and that's great". Just make yourself a bit happy. Sometimes even a fake smile is enough to improve your mood.)
If you happen to catch yourself before you bite your finger (but already moving it towards your mouth, or somehow preparing to do it), congratulate yourself for noticing that, and put the hand down, and congratulate yourself for putting your hand down, because this is what you want to achieve. Never happens spontaneously? Then make it happen!
Do this exercise: just sit (like you usually do when it happens), raise your hand halfways to your mouth, now stop and realize that at this moment you have two options. (You can imagine two colored arrows from your finger; one towards your mouth, the other one towards the desk.) Choose the better option and put your hand down. Congratulate yourself for making the right choice, and feel happy for a moment. Now do this again. And again. And again. You are now building a habit that can increase the probability of catching yourself in the real situation.