In short
The phrase at that time
refers to when the report of Jesus’ miracles reached Herod, some time after John’s death—the period described in the previous passage (Matthew 13:54–58).
The phrase at that time
used in Matthew 14:1 refers to the moment when the report of Jesus’ miracles reached Herod. We know that this was after the death of John (see Matthew 14:2).1
These opening words of Matthew 14:1–36 point to the period when people throughout Palestine were becoming aware of the miracles performed by the Saviour—the same period that had been described in the preceding passage (Matthew 13:54–58).2
1 At that time Herod the tetrarch heard about the fame of Jesus,