The root verse the sources return to is from Micah, and it asks for justice, mercy and walking modestly with God. Nothing in it is about clothing. In the classical rabbinic material a tzanua is a kind of person: someone who does not announce their own charity, who does not press their opinion into every room, who conducts private matters privately. The Talmud praises modesty in men explicitly, and several of its examples concern the behaviour of scholars rather than of women.
The value governs speech as much as dress. Boasting, gossip and the public airing of a family difficulty are failures of tzniut, and so is display of wealth. The rabbinic texts on hospitality and on entering a house treat privacy as something owed to other people, which is the same value read from the other side.
The move from disposition to detailed regulation happens in the halakhic literature, where tzniut becomes a heading under which the codes place rules about dress, about which parts of the body are to be covered, about the separation of men and women at gatherings, and about a man reciting prayer within sight or hearing of things that would distract him. Maimonides and later Joseph Karo organise the material, and the later responsa literature is where the detailed contemporary standards were worked out, community by community.