If you are an espresso aficionado, you should try to hit Intelligentsia during your trip. There are three locations (addresses are below). The Millennium Park location is the closest to the Hancock, though it is a mile away.
I believe there is a Lavazza in the Drake Hotel (140 E Walton Pl), which is about a block from the Hancock. That is a safe bet.
Lakeview:
3123 North Broadway, Chicago, IL
(773) 348-8058
Across the street from Millennium Park
53 E Randolph St, Chicago, Illinois
(312) 920-933
In the Monadnock Building in the Loop
53 West Jackson Boulevard, Chicago, IL
(312) 253-0594