Gender bias is a significant issue in machine translation, leading to ongoing research efforts in developing bias mitigation techniques. However, most works focus on debiasing of bilingual models without consideration for multilingual systems. In this paper, we specifically target the unambiguous gender bias issue of multilingual machine translation models and propose a new mitigation method based on a novel perspective on the problem. We hypothesize that the gender bias in unambiguous settings is due to the lack of gender information encoded into the non-explicit gender words and devise a scheme to encode correct gender information into their latent embeddings. Specifically, we employ Gender-Aware Contrastive Learning, GACL, based on gender pseudo-labels to encode gender information on the encoder embeddings. Our method is target-language-agnostic and applicable to already trained multilingual machine translation models through post-fine-tuning. Through multilingual evaluation, we show that our approach improves gender accuracy by a wide margin without hampering translation performance. We also observe that incorporated gender information transfers and benefits other target languages regarding gender accuracy. Finally, we demonstrate that our method is applicable and beneficial to models of various sizes.