The city moving the farthest up the top ten rankings between 1850 and 1880 was

In population rankings of cities in the United States, Chicago moved up the furthest in the top ten rankings between 1850 and 1880. 

In 1850, Chicago wasn't even in the top ten list.  Chicago's population in 1850 was only around 30,000.  But by 1880, the city of Chicago had expanded rapidly, to a population of over 500,000 and into the #3 spot on the list of America's largest cities.   (On some lists, you might see Chicago as #4 on the list at that time, but that's if they count New York and Brooklyn separately rather than counting the New York metropolitan area as a unit.)