You could look into semi-supervised learning, which is useful for training models when you have both labeled and unlabeled data. Semi-supervised methods consider the distribution of unlabeled data to improve the performance of your model. The following picture should give you some intuition regarding how unlabeled data can be useful.
In another direction, you may train a classifier with the labels you have so far. Then, use the classifier to predict the probability of each label for your unlabeled data. Sort labels by their probability, and manually label a small sample of low (p<0.25), medium (0.25 < p < 0.75) and high (p> 0.75) probabilities. Then, try to estimate in which probability range your model is struggling most. In theory, it should be a better investment of your time to manually label the cases that fall in the medium probability range, as these are the ones your current model is more uncertain about. This and similar approaches belong to the category of active learning.
In short, look into semi-supervised or active learning.