That must be a nuisance! We have cards for Union County (where we live) and Mecklenburg County, which has much more stuff.
I did finally acquire a Queens library card this past year, as I was working on Paul's congressional campaign, because a large chunk of his district is located in neighborhoods like Maspeth, Ridgewood, Sunnyside, Woodside, etc., etc...
I have to admit, though, they have a much better selection there-at least among local branches-than their counterpart in my borough.