We study how to design learning-based adaptive controllers that enable fast and accurate online adaptation in changing environments. In these settings, learning is typically done during an initial (offline) design phase, where the vehicle is exposed to different environmental conditions and disturbances (e.g., a drone exposed to different winds) to collect training data. Our work is motivated by the observation that real-world disturbances fall into two categories: 1) those that can be directly monitored or controlled during training, which we call "manageable", and 2) those that cannot be directly measured or controlled (e.g., nominal model mismatch, air plate effects, and unpredictable wind), which we call "latent". Imprecise modeling of these effects can result in degraded control performance, particularly when latent disturbances continuously vary. This paper presents the Hierarchical Meta-learning-based Adaptive Controller (HMAC) to learn and adapt to such multi-source disturbances. Within HMAC, we develop two techniques: 1) Hierarchical Iterative Learning, which jointly trains representations to caption the various sources of disturbances, and 2) Smoothed Streaming Meta-Learning, which learns to capture the evolving structure of latent disturbances over time (in addition to standard meta-learning on the manageable disturbances). Experimental results demonstrate that HMAC exhibits more precise and rapid adaptation to multi-source disturbances than other adaptive controllers.