Mark, it's not really much of a "neighborhood" in the usual sense, the area is dominated by the vast convention center McCormick Place.
However, you can easily get downtown by bus from there. The "twin" of the Saxophone Shop, something like Band and Orchestra Company, used to be in the DePaul Center, but it's been out of business for a while now. As far as I know there's not a whole lot interesting in the way of music stores downtown anymore.
If you are ambitious you could ride the el to Evanston to visit Paul Maslin, pmwoodwind.com. He's got a large selection of used Selmer saxophones to try and does excellent repair work if you are bringing a horn along (but call him first to let him know you are coming if you want him to look at your horn).
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