Anyone have experience with changing tire size to improve handling?

My first new car, a 1987 Ford Escort, came with 165/80R/13 tires, which I now think was inexcusable. I eventually changed them out for 185/70R13 tires and the handling for things like evasive maneuvering was noticeably improved. Your 185/60R/14 tires shouldn’t be that terrible, though. The reason I mention evasive maneuvering is that when the Escort still had the original tires, I was going about 20 MPH toward an intersection and a drunk turned left in front of me. When I tried to brake and steer to avoid said drunk (this happened on good, dry pavement) the car slid almost like it was on ice.