Fine-tuned Bidirectional Encoder Representations from Transformers (BERT)-based sequence classification models have proven to be effective for detecting Alzheimer's Disease (AD) from transcripts of human speech. However, previous research shows it is possible to improve BERT's performance on various tasks by augmenting the model with additional information. In this work, we use probing tasks as introspection techniques to identify linguistic information not well-represented in various layers of BERT, but important for the AD detection task. We supplement these linguistic features in which representations from BERT are found to be insufficient with hand-crafted features externally, and show that jointly fine-tuning BERT in combination with these features improves the performance of AD classification by upto 5\% over fine-tuned BERT alone.