Online social media provides a channel for monitoring people's social behaviors from which to infer and detect their mental distresses. During the COVID-19 pandemic, online social networks were increasingly used to express opinions, views, and moods due to the restrictions on physical activities and in-person meetings, leading to a significant amount of diverse user-generated social media content. This offers a unique opportunity to examine how COVID-19 changed global behaviors regarding its ramifications on mental well-being. In this article, we surveyed the literature on social media analysis for the detection of mental distress, with a special emphasis on the studies published since the COVID-19 outbreak. We analyze relevant research and its characteristics and propose new approaches to organizing the large amount of studies arising from this emerging research area, thus drawing new views, insights, and knowledge for interested communities. Specifically, we first classify the studies in terms of feature extraction types, language usage patterns, aesthetic preferences, and online behaviors. We then explored various methods (including machine learning and deep learning techniques) for detecting mental health problems. Building upon the in-depth review, we present our findings and discuss future research directions and niche areas in detecting mental health problems using social media data. We also elaborate on the challenges of this fast-growing research area, such as technical issues in deploying such systems at scale as well as privacy and ethical concerns.