Traumatic Brain Injury (TBI) is caused by a head injury that affects the brain, impairing cognitive and communication function and resulting in speech and language disorders. Over 80,000 individuals in the US suffer from long-term TBI disabilities and continuous monitoring after TBI is essential to facilitate rehabilitation and prevent regression. Prior work has demonstrated the feasibility of TBI monitoring from speech by leveraging advancements in Artificial Intelligence (AI) and speech processing technology. However, much of prior work explored TBI detection using scripted speech tasks such as diadochokinesis tests or reading a passage. Such scripted approaches require active user involvement that significantly burdens participants. Moreover, they are episodic, are not realistic, and do not provide a longitudinal picture of the user's TBI condition. This study proposes a continuous TBI monitoring from changes in acoustic features of spontaneous speech collected passively using the smartphone. Low-level acoustic features are extracted using parametrized Sinc filters (pSinc) that are then classified TBI (yes/no) using a cascading Gated Recurrent Unit (cGRU). The cGRU model utilizes a cell gate unit in the GRU to store and incorporate each individual's prediction history as prior knowledge into the model. In rigorous evaluation, our proposed method outperformed prior TBI classification methods on conversational speech recorded during patient-therapist discourses following TBI, achieving 83.87% balanced accuracy. Furthermore, unique words that are important in TBI prediction were identified using SHapley Additive exPlanations (SHAP). A correlation was also found between features acquired by the proposed method and coordination deficits following TBI.