Richard Hofstadter's "The Age of Reform: From Bryan to F.D.R."(1955) may be a good place to start. It covers the period from 1890-1941, linking Progressivism to the later New Deal reforms. Hofstadter writes really well and attempts to paint the appeals of the different reform movements of the period to ordinary people as well as the intellectual debt the New Deal owed to the Progressive Era.

If you don't want to read the book but want to get a sense of the debate it engendered, try Alan Brinkley's "The Age of Reform: A Reconsideration" (1985)

The Progressive Era is not a reference to a period of political liberalism, but to a period in which political discourse was largely dominated by the notion that the United States was "progressing" towards something. Lots of people had different ideas of what exactly the end point of all this progression would be. For instance, there were those who Americans today might identify with the modern use of the word "progressive", in that they supported workers rights, women's rights, opposed monopolies, and favored social programs and taxes designed to redistribute wealth. There were also prohibitionists, who saw America progressing towards a more sober future. There were those who saw the need to make America a gleaming city on a hill for the entire world to envy, and those who saw the United States as the future leader of the world, exporting our values abroad.

While the Progressive Era certainly encompassed reforms like settlement houses and reformers like Jacob Riis that were concerned about tenement life and the plight of the poor and working class, a lot of the real movement and change was directed at causes that benefitted the middle class.

As standards of living and expectations about quality of life rose, middle class families were able to be interested in and advocate for sanitary and safe food products, cleaner cities with parks and other amenities, more transparent and effective medical care, and other quality of life issues.

While it's certainly true that any individual member of the middle class certainly had less power and influence than a Gilded Age robber baron, the middle class that had power in numbers. As much as the deck was stacked against them, and as corrupt as the political system can be (and was), the democratic process could still produce change and bring Progressive politicians representing middle class interests to power.

This is, of course, not to say it worked perfectly. Plenty of Progressive reforms were limited in their ability to address social/economic problems or even made them worse (both intentionally and unintentionally).