Look over the speaker carefully. I've repaired old Black Widows that had glue seams let go. Over time the aluminum frame may develop white "corrosion" causing the glue to let go. I've seen both the edges at the frame let go, and the corrugated spider glue. Clean up the aluminum the best you can and reglue with the best you have. (I don't remember what I used...I've been out of the repair business for 10 years.) The corrosion happens due to dampness and condensation, like say from storing cabinets in the garage.
Another thing is to take off the basket, and inspect the coil. I have worked on aluminum forms that had developed high spots (from heat) and have worked out the high spot with the plastic grip of a screwdriver. I have worked on ones that I just couldn't get to stop rubbing.