Causal mediation analysis is a popular approach for investigating whether the effect of an exposure on an outcome is through a mediator to better understand the underlying causal mechanism. In recent literature, mediation analysis with multiple mediators has been proposed for continuous and dichotomous outcomes. In contrast, methods for mediation analysis for an ordinal outcome are still underdeveloped. In this paper, we first review mediation analysis methods with a continuous mediator for an ordinal outcome and then develop mediation analysis with a binary mediator for an ordinal outcome. We further consider multiple mediators for an ordinal outcome in the counterfactual framework and provide identification assumptions for identifying the mediation effects. Under the identification assumptions, we propose a regression-based method to estimate the mediation effects through multiple mediators while allowing the presence of exposure-mediator interactions. The closed-form expressions of mediation effects are also obtained for three scenarios: multiple continuous mediators, multiple binary mediators, and multiple mixed mediators. We conduct simulation studies to assess the finite sample performance of our new methods and present the biases, standard errors, and confidence intervals to demonstrate that our proposed estimators perform well in a wide range of practical settings. Finally, we apply our proposed methods to assess the mediation effects of candidate DNA methylation CpG sites in the causal pathway from socioeconomic index to body mass index.