No. Context is solely focused on the urban fabric of Istanbul itself, and our walks—at the moment—take place solely in the city. In terms of a day trip, Bodrum is the closest historic site. Ephesus, Troy, and the other major sites on the Aegean are quite a distance from Istanbul. We work with an exceptional tour operator in southern Turkey who organizes multi-day tours to these sites (and many others) called Equinox. It is founded by an archaeologist and has the same love of place as Context. I'm interested in either a full-day or two half-day walks. What do you suggest? Our general advice is to split a full-day walk into two half-days. It gives you a chance to absorb more.
In terms of the main sights, we usually approach the Hagia Sofia and Topkapi Palace as two separate walks—the Hagia Sofia Seminar and the Topkapi Palace walks, respectively. Partly this is because there is such a wealth of material to cover with each. But, also, because each site refers to a very distinct historical period: the Byzantine empire with the Hagia Sofia, and the Ottoman Empire with Topkapi. What many people do is choose one for the morning, and then explore the other in the afternoon on their own.
In terms of getting off the beaten path, I have two recommendations. The first is our History and Continuity in Istanbul walk, which Travel + Leisure called one of the best walking tours in Europe. It includes the Chora Church (late Byzantine) but also a variety of other historical time periods, including late Roman, Ottoman, and modern. It includes a walk through Balat, historically home to a very diverse population of Jews, Christians, and conservative muslims. And, so, the walk also looks at the changing face of the city and its multiculturalism. The second walk I would suggest is Cosmopolis, which looks at the diplomatic quarter of Galata, Pera, and Istiklal Street. This walk also focuses on diverse populations in Istanbul and the period of expansion and liberalization of Ottoman rule, from the 17th century to 20th. Parts of this neighborhood are very cutting edge and bohemian today.