Apart from poor Millard, whom I didn't expect to make the new Atlantic list, several others immediately spring to mind as worthy of consideration. In more or less chronological order:
Andrew Jackson may have been the figurehead, but it was Martin Van Buren who invented the Democratic Party and the Second Party System, then abandoned the party he had created to serve as the presidential candidate for the first non-trivial anti-slavery party, the precursor to the Republican Party.
Daniel Webster's stirring oratory focused and inspired among a generation of Americans unionist sentiment that they would rise to defend when the crisis came.
Abolitionists get all the press, but it was David Wilmot who rose in the House in 1846 to move the famous Proviso that set the country on the road to Civil War.
After crafting and pushing through Congress the Compromise legislation of 1850 that averted civil war for ten years, Stephen A. Douglas's decisions to introduce the Kansas-Nebraska Act and later oppose the Lecompton Constitution gave birth to the Republican Party and placed it on the road to victory in 1860, sparking the war he sought to avoid.
Ann Althouse's musings on the list are worth your time.
About the illustration, entitled Marriage of the Free Soil and Liberty Parties (1848):
A comic portrayal of the alliance between Free Soil Democrats and Whigs and the more extremist abolitionist Liberty party interests during the election campaign of 1848. The factions joined to form the Free Soil party and nominated a presidential candidate in a convention at Buffalo in August. That union is lampooned here as the wedding of Free Soil presidential candidate Martin Van Buren (center left) and a ragged black woman (center right). Van Buren ally Benjamin F. Butler presides over the "marriage." Van Buren, reluctant to embrace the aged bride, is shoved forward by antislavery editor Horace Greeley (left), who says, "Go, Matty, and kiss the bride That is an indispensable part of the ceremony." Van Buren's son John (far left, here called "John Van Barnburner") also urges him on, "Walk up, dad. You can hold your breath till the ceremony is over, and after that you can do what you please." Van Buren says, "I find that politics, as well as poverty, make one acquainted with strange bedfellows." In contrast, the woman beckons with open arms, "Come here, my flower. You is a great stranger, and I want to get acquainted wid you." A black man behind her says of Van Buren, "I nebber hab berry good pinion ob the gemman; but if he ax pardon for all he hab done and said agin us, I will shake hands wid de genman." A black woman (further right) remarks, "Mercy on me! How bashful he is!" Butler, with arms raised and book in one hand, intones, "Who giveth this man to be married to this woman?"