Hi,
This is an excellent question, and it's something that irks me as well. If you listen closely on many records, you'll hear that many bass players do have this 'problem', if you want to call it that. You can even hear it on double bass recordings if the player is playing pizzicato (with fingers, not bow).
For the most part, it will not be noticeable in a band setting because it isn't very loud, and I think most people responding to this thread will say to simply not worry about it.
Being the perfectionist that I am I have put some (but not nearly enough!) effort into getting rid of it. Here's what I have come up with:
- Release the note 'faster'. That is, do not gradually release the pressure on the note but do it suddenly when you want your note to stop. This will eliminate the critical window of time between having a 'fully fretted' and a 'fully muted and released' string, thus minimizing the time in which your problem can occur.
- Use all of your other fingers to mute the string
just before taking the string off the fret. So if you just played a note with your index, when you want the note to end you would lightly touch the string with your middle, ring, and pinky fingers before releasing the index. You have to practice it because if you come down too hard with those fingers the string will make contact with the frets, and you'll end up making more noise than if you just released in the normal way

- The above method is extremely difficult to do well - it requires a lot of coordination, but usually the movements have to be done rapidly for it to sound seemless. It is especially hard to do when you're fretting with the ring because you need a good deal of independence between ring and pinky if you want to fret with ring and mute with pinky. And, of course, it is impossible to do when fretting with pinky, because, well, you don't have any more fingers after that!
I guess the
real solution to try here would be right hand muting, or some combination of effects and methods, such as using a sponge or leather dampener at the bridge, EQ, using more neck pickup, compression, etc
I haven't really tried any of these.
So, hopefully someone else can provide a more useful answer and help us both out!