What are some other words for "alley" (that place behind your garage (car port) that one keeps the trash cans (rubbish receptacles) here in the "Very Large Northeastern Town in Illinois"?

I don't know about Illinois but I can give you some Canadian (Manitoba) alternatives. I would call it a "back lane", where people keep their "garbage cans". Also, for me, a garage has a door and is completely enclosed, whereas a carport is sort of a roof on legs, with some walls, or no walls, but definitely not enclosed.

However - now that I've moved the "the oldest city in North America" I have come to realize that back lanes and garages are relatively new things. We live in an older part of St. John's so there are no back lanes - everyone's yard abuts everyone else's. Everyone parks on the street, unless they've paved their front "lawn" (postage stamp size) into a parking pad. And no one has garbage cans because the wind would make the neighbourhood into a very dangerous place on garbage day!