Yes. The Gemara calls this area a beit sha’ar, literally a gate-house. Although it does not meet the minimum size of a living space, since it serves as a passage into a living area, it needs a mezuzah without a berachah.
Commonly, people like to put their halachically best and aesthetically most beautiful mezuzah on their front door. In your case, you should rather put that one on what you call the “real” front door, which is fully obligated with a berachah. Then, you could proceed to place a mezuzah on the outer door.
 Kuntres HaMezuzah 286:152; Agur B’ohalecha 36:37; Teshuvos Maharsham 3:154.
 Agur B’ohalecha 1:54.