Mental health in athletes is shaped by both interpersonal and intrapersonal psychological resources. This study examined whether the association between athletes’ perceptions of the coach–athlete relationship and emotional, psychological, and social well-being showed statistically significant indirect associations through competitive state self-confidence. A cross-sectional sample of 211 adult Hungarian athletes (149 women and 62 men; mean age = 23.39 years, SD = 3.58) completed validated Hungarian measures of the coach–athlete relationship, competitive state self-confidence, and positive mental health. Participants were recruited online by distributing the questionnaire directly to athletes and through sports clubs. Three separate single-mediator models were estimated. Athletes’ perceptions of the coach–athlete relationship were positively associated with competitive state self-confidence (β = 0.41, p < 0.001), which was positively associated with emotional (β = 0.26, p < 0.001), psychological (β = 0.30, p < 0.001), and social well-being (β = 0.19, p = 0.010). Significant indirect associations through competitive state self-confidence were observed for emotional (β = 0.11, p = 0.002), psychological (β = 0.12, p < 0.001), and social well-being (β = 0.08, p = 0.016), whereas the corresponding direct associations were not statistically significant. These findings indicate that the observed associations were statistically consistent with indirect associations through competitive state self-confidence. However, the cross-sectional design does not establish temporal ordering or a causal mediating mechanism. Longitudinal and intervention-based research is needed to determine the temporal direction of these associations.