Part of the problem, of course, is that Latin America is full of people very different from the white Anglo protestant USA. I see two possibilities. Either the US goes all Decade of Darkness and subjugates most of the locals as unfree non-citizens, or it goes massively multicultural and treats brown, black and Indio the same as whites. Which seems unlikely.
Your best bet is to have an existential threat in the Old World (like otls Nazis or Soviets, but who manage to conquer the whole continent). In that case the US might well find the will to 1) industrialize and help educate, and then 2) bring into its political structure, the rest of the Americas. That would, of course, mostly not be 'conquering', but more likely accepting applications to join.
Second best bet is for the US to annex all of Mexico after a different Mexican-American war. The US formalizes voting tests, e.g. requiring literacy and/or a certain income to be able to vote, probably as well as an oath of allegiance to the Constitution, say. After a generation of nasty politics, the whiter elite of Mexico is co-opted into the US political system, and several 'brown' politicians (initially mediterranean dark, but later mestizo) and businessmen become prominent in US political and financial circles.
The large mass of nonCitizen darker, poorer Hispanics makes Southern slavery seem less exceptional, and, even if slavery, per se, doesnt massively extend into Mexico, nonCitizen subjects are far too numerous to be allowed the vote, and the threat to the Southron way of life eases. Eventually, slaves move from 'property' to peons/indentured labour.
Meanwhile, the increasing 'browner' cast to the American elite means the 'one drop' rule, firstly becomes unenforceable, and later is removed.
By the 1880s, things have stabilized, and border disputes, etc, in Central America end with each of those states being conquered piecemeal, and treated the same way as Mexico.
In 1892, Ecuador, in a border dispute with Colombia, applies voluntarily for admission to the Union as a State. Colombia refuses to accept this, and is soon conquered by the US. So, for a decade or so, Ecuador is the only State in South America, Colombia is just a territory.
Border disputes with Peru and Venezuela lead to those being conquered, while Chile leaps in on thePeru war, and gains 'favored status'.
Next, Paraguay seeks entrance due to threats from HER neighbors.
Argentina and Brazil go to war, and Uruguay petitions of admittance to the Union to avoid being hit in the crossfire. The US imposes a peace, reducing both Argentina and Brazil to protectorates.
Chile, seeing which way the wind is blowing, and wanting not to slide/be forced into protectorate status, petitions to become a State.
WWI equivalent happens, and the US occupies the 3 Guyanas, 'to keep order'. The European nations arent in any kind of position after the war to object, especially as the US presents them with a bill for 'the cost of peacekeeping and administration' incurred.
WWII equivalent happens, and Argentina and Brazil continue their rivalry by aiding the opposite sides. Since the US has limited their militaries, this largely consists providing safe harbor for European warships and raiding each other.
the US intervenes, kicks out all Europeans, and forcibly annexes the last two surviving South American countries. Then mops up the Caribbean, any bits that remain, just for completeness.
By about 1950 or so, the US owns all of Latin America and has I dont know how many states, but probably over 100. In states like Dakota, all residents of the state are citizens (but non-citizen subjects of other states, while allowed in for migrant labor, face major difficulty achieving resident status). In states like Alabama and the Yucatan, less that half the population holds citizenship. In the newly aquired territories in exBrazil, only about 1% (mostly american collaborators and carpet baggers), but that number is expected to go up 'soon'.
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Kind of Decades of Darkness lite.