Train to avoid false negatives
What your network learns depends on the loss function you pass it. By choosing this function you can emphasize various things - overall accuracy, avoiding false negatives, false positives etc.
In your case you probably use a cross entropy loss in combination with a softmax classifier. While softmax squashes the prediction values to be 1 when combined across all classes, the cross entropy loss will penalise the distance between the actual ground truth and the prediction. In this calculation it will not take into account what the values of the "false negative" predictions are. In other words: The loss function only cares for the correct class and its related prediction, not for the values of all other classes.
Since you want to avoid false negatives this behaviour is probably the exact thing you need. But if you also want the distance between the actual class and the false predictions another loss function that also takes into account the false values might even serve you better. Give your high accuracy this poses the risk that your overall performance will drop.
What to do then?
Making the wrong prediction and being very sure about it is not uncommon. There are millions of things you could look at, so your best guess probably is to investigate the error. E.g. you could use a confusion matrix to recognize patterns which classes are mixed with which. If there is structure you might need more samples of a certain class or there are probably labelling errors in your training data.
Another way to go ahead would be to manually look at all (or some) examples of errors. Something very basic as listing the errors in a table and trying to find specific characteristics can guide you towards what you need to do. E.g. it would be understandable if your network usually gets the "difficult" examples wrong. But maybe there is some other clear systematic your network did not pick up yet due to lack of data?