An extended walk is an A to B walk of more than a day or two.
Your primary decision is whether you prefer mountains or plains. This will dictate your choice of east or west.
Your other essential choice is which dynasty's wall you want to walk; Ming, Han, or Qin, for example.
If you want to walk in the mountains, you will be choosing between Hebei Province and Beijing Municipality. If you can devote the extra time to the travel, Hebei can be more rewarding than Beijing due to its smaller number of residents and tourists plus its very impressive Great Wall locations. It offers many choices. In eastern Hebei Province (to the east of Beijing), you could start at Laolongtou at the sea and walk west from there. This does present some serious danger in the Jiaoshan to Sandaoguan areas. You can also choose from a number of other places, especially in Qinhuangdao. Many different possibilities are available in the Lengkou area alone. In addition, there are many possibilities for long walks in the Zhangjiakou area.
If you prefer the Beijing area, you will choose between the walls in Huairou District and the wall that runs along the Beijing-Hebei border west of Badaling. Both of these are challenging and difficult options. In Huairou, you could start at Shentangyu and proceed to the west from there. In Yanqing, you could start near Badaling and head southwest, or you could start in the Shuitoucun area at Bijiashan and head to the northeast. There are other choices in Beijing, but these are probably the least difficult.
Whatever place you choose in the mountains, a long hike along the Great Wall in the mountains will prove to be a formidable goal with inherent risks. Also, be prepared for the fact that certain Great Wall sections will be unsafe or impossible for one reason or another. Frequently, some way around the problem area will have to be found.
If you don't like the idea of a long walk in steep and difficult mountains, there are alternatives. As the wall extends westward, the terrain gradually becomes more manageable. This gives you the opportunity to select the type of terrain you want, from more gentle hills to essentially level plains. And you also get the opportunity to experience the parts of the Great Wall that are not commonly visited by tourists, the walls whose materials and construction differ from the brick and stone walls that people usually envision when talking about the Great Wall of China. And in general, the further west you go, the tamer the terrain becomes, even though the elevation gets higher. Shanxi and Shaanxi are less mountainous than Beijing and Hebei. Ningxia is even gentler, and Gansu is the flattest.
Most people will select the Ming Dynasty Great Wall, and it offers a limitless variety of possibilities for an extended walk. But consider walking on a different dynasty's wall, especially if you have a particular scientific or historic interest, or if you already have some experience with the Ming wall. The Han Dynasty, Qin Dynasty, or Jin State wall would be the most likely other choices. In some areas, you could combine segments from different dynasties to create a memorable extended walk.
The Datong area provides many opportunities for extended walks that can include walls from a variety of different building periods.
If you decide on a walk along the Ming Dynasty Great Wall to the west of Beijing and Hebei, consider hiking a portion that is not on the usual thru-hiking path between Shanhaiguan and Jiayuguan. For example, all or a portion of the inner Shanxi Great Wall offers an exciting opportunity to walk on a rarely visited stretch of the wall.