The discrete choice paradigm of random regret minimization (RRM) has been recently proposed in several choice contexts. In the route choice context, the paradigm has been used to model the choice among three routes and to formulate regret-based stochastic user equilibrium. However, in the same context the RRM literature has not confronted three major challenges: (i) accounting for similarities across alternative routes, (ii) analyzing choice set composition effects on choice probabilities, and (iii) comparing RRMbased models with advanced RUM-based models. This paper looks into RRM-based route choice models from these three perspectives by (i) proposing utility-based and regret-based correction terms to account for similarities across alternatives, (ii) analyzing the variation of choice set probabilities with the choice set composition, and (iii) comparing RRM-based route choice models with C-Logit, Path Size Logit and Paired Combinatorial Logit. The results illustrate the definition of the correction terms within the regret function, the effect of the choice set specificity of RRM-based route choice models, and the positive performance of these models when compared to advanced RUM-based models.