In a lot of cases, you deal with too many features and you either try to reduce the dimensionality of your feature space with techniques like PCA or you try to select only the relevant ones. The latter is known as feature selection. However, what you have is a situation where you don't have a lot of features and the information you get from those does not seem to be enough for a solid classification.
Generally speaking, the best options you have in this case are cleaning your data as good as possible and gathering more features. This can either be done by using new sources of information or by deriving new information from existing sources.
Usually, you derive the best feature candidates for your model from the literature of the respective subject area. Domain knowledge helps a lot!
In your case, I would guess that variables like time of day, weekday and day of the month could provide some valuable information, apart from the mail content of course. But these are simple guesses. In the end, you will have to dig into the literature and find the right features by trial and error.
Since you are trying to classify incoming emails, I assume you have access to the mail subject and the mail body. In this case, I would recommend starting with a simple Naive Bayes algorithm, since it is easy to use, effective and there are many implementations and guides how to use them. Also, Naive Bayes can easily be applied to multiclass problems like yours.