This may be difficult to visualize, but, after Dec. 21st, the sunrise should get earlier and sunset later, both extending the amount of daylight. As you point out, while the sun does begin to set later, for several days after the 21st the sunrise also gets later.
This curious phenomenon occurs because we rely on watch time rather than sundial time, and because after the winter solstice, it starts taking longer for the sun to make it from one east-west point in the sky back to that same point.
The Earth takes 23 hours and 56 minutes to spin around on its axis. But the average day is 24 hours long because as the earth is spinning, it is also revolving around the sun. The spin makes the sun appear to move from east to west. The revolution makes the sun appear to move from west to east. So for the sun to make it back to the same east-west point in the sky, the Earth needs to spin a little bit longer than one rotation � for about four more minutes.
Therefore, anything that increases the west to east motion will also increase the length of day (the Earth will need to spin a bit longer for the sun to get back to the same point in the sky). After the solstice, both the tilt of the earth on its axis and the variability in the speed of Earth's revolution around the sun increase the sun's west-to-east motion.
(Contd.)