Hallucinations remain one of the major barriers to the reliable deployment of Large Language Models (LLMs). Recent works have explored hallucination detection using both supervised classification based approaches and unsupervised metric based approaches, with the latter remaining popular since they do not require labeled data. In this work, we study the performance and generalizability of these approaches through a comprehensive empirical analyses across 11 datasets and 10 models. We first establish that supervised methods achieve stronger in-domain performance but lag behind unsupervised methods in the out-of-domain setting. Then to narrow this generalization gap, we introduce a simple approach that improves the generalizability of supervised hallucination detectors by training on a curated, multi-domain data mix, which can complement the subsequent addition of task-specific data. In our experiments on hallucination detection on 697K QA samples from 12 open source QA datasets, we show that incorporating this general training allows supervised methods to surpass unsupervised metric based methods by an average of +7.25% on out-of-distribution data within the short-form QA setting, without addition of any task-specific data. We also analyze scaling behaviors and estimate how much task-specific data is required to achieve reliable performance, finding that models augmented with general data require up to 40.3% less task-specific data to achieve close to optimal performance. Together, our findings highlight a simple path toward fortifying supervised hallucination detectors against domain shift.