Perhaps you could add some sort of hysteresis such that it continues to report you as being in the previous grid square unless you go >1/2 a grid square distance away from it
Changing what you use as the boundary doesn't change the fact that if you're close enough to the actual boundary, you will jump a lot. So you have to go quite large with the boundary for it to limit the pinpointing. Having larger areas within your boundaries makes the feature much less 'useful' though.
Which granularity: Harlem/Hell's Kitchen etc? West Harlem/East Harlem etc? Manhattan/Long Island? New York/New Jersey etc.?
Feature wise you would probably want at least something like Harlem/Hell's kitchen granularity and there are unfortunately enough people living on the borders of all of these that you could pinpoint those just from GPS inaccuracies.
If you lived on a boundary it would be very clear because your location would change very often, perhaps just by walking to the kitchen.