What is the ruling on a supererogatory fast for one who has not heard the adhan and the night has not yet fully departed?

The fast begins by abstaining from things that break the fast at the advent of true dawn, not by hearing the adhan, unless the adhan coincides with the dawn. This is based on the Almighty's saying: "And eat and drink until the white thread of dawn becomes distinct to you from the black thread [of night]. Then complete the fast until the night." (Al-Baqarah: 187).

Accordingly, your fast is valid if you abstain from things that break the fast before the advent of true dawn, and also if you are in doubt about the advent of dawn.