the sixth hour of the morning; around 6 a.m. (in the traditional Japanese time system)
Refers to the sixth 'hour' (六つ) of the day in the old Japanese temporal hour system, which began at dawn. 明け六つ specifically marks the hour around sunrise, roughly equivalent to 6 a.m. in modern time. Used historically; now encountered mainly in literature, historical contexts, or explanations of the old time system.
In the Edo period, the bell rang at the sixth hour of the morning.
Around the sixth hour of the morning, the traveler left the inn.