To understand the extent of American expansionism, it's important to understand what motivated it in the first place. John Mearsheimer explains it as a drive for regional hegemony. When the United States gained its independence, it was "a relatively weak country whose people were largely confined to the Atlantic seaboard" and surrounded by European powers and the Native Americans to the West.

Over the next seven decades, the Americans responded to this precarious situation by marching across their continent to the Pacific Ocean, creating a huge and powerful country in the process. To realize their so-called Manifest Destiny, they murdered large numbers of Native Americans and stole their land, bought Florida from Spain (1819) and what is now the center of the United States from France (1803). They annexed Texas in 1845 and then went to war with Mexico in 1846, taking what is today the American southwest from their defeated foe. They cut a deal with Britain to gain the Pacific northwest in 1846 and finally, in 1853, acquired additional territory from Mexico with the Gadsden Purchase.

The plain truth is that in the nineteenth century the supposedly peace-loving United States compiled a record of territorial aggrandizement that has few parallels in recorded history. It is not surprising that Adolf Hitler frequently referred to America’s westward expansion as a model after Germany invaded the Soviet Union in June 1941. “Here in the East,” he said, “a similar process will repeat itself for a second time as in the conquest of America.”

By the latter half of the 19th century, the United States was the major power in the Western hemisphere. The European powers had been relegated to minor roles, and were no longer threats to American regional hegemony. While consideration was given to conquering further territory, the project of American expansion had firmly achieved its goal once the United States was secure in the sense that its power couldn't be challenged by any other player in the region.

While the United States likely could have expanded further south, Mexico following its civil strife of the 19th century was not a serious threat the American power.

After this point, Mearsheimer argues, the country began to look farther afield, to prevent great powers in other regions from becoming hegemonic and therefore able to project power (Germany in Europe, Japan in the Pacific, and finally Russia in Eurasia).

In The Half Has Never Been Told, E Baptist talked about this. Pre Civil War, slavery (and the cotton industry) depending on expansion of slavery and cotton growing. Some politicans in the South wanted to expand into Cuba, even going as far as offering $100 million to buy it (which was reject by Spain). Some wanted to expand west to California, some wanted political capital spent on trying to make slavery (essentially) legal throughout the USA. But the North didn't want any more slave states, which stopped that sort of expansion. And then the Civil War ended slavery.

There were indeed calls at the time of the Mexican-American war for the United States to take over all of Mexico, referred to as the "All Mexico" movement. This cause was led by Democrats from the major urban centers along the eastern seaboard of the United States, and found major purchase in summer 1847 following a string of American successes on the battlefield (notably the Battle of Chapultepec that culminated in the capture of Mexico City).

As to why it didn't happen, there's a few reasons. One reason is simply racism. Senator John C. Calhoun said in a speech in early 1848 on the Senate floor:

We have never dreamt of incorporating into our Union any but the Caucasian race—the free white race. To incorporate Mexico, would be the very first instance of the kind, of incorporating an Indian race; for more than half of the Mexicans are Indians, and the other is composed chiefly of mixed tribes. I protest against such a union as that!

But, there are several pragmatic reasons as well. Pacifying Mexico would have been a daunting task. While the southwest territory that America gained comprised 40% of Mexico's land, it was sparsely populated with only about 80,000 total people. By contrast, the rest of Mexico held 8,000,000.

Ultimately though, Frederick Merk concludes in his book "Manifest Destiny and Mission in American History: A Reinterpretation" that the territorial expansion of the USA was just never popular to begin with. The Mexican-American war was opposed from the start by the Whig Party, with then Rep. Abraham Lincoln being a particular vociferous opponent of it. And although its major proponents were Democrats, there were many in the party who had more mixed feelings (Calhoun being a good example of a Democratic Senator who supported the annexation of Texas but opposed conquering all of Mexico). In such a political environment, adding Mexico to the Union would have been untenable.

Sources:

"Manifest Destiny and Mission in American History: A Reinterpretation" by Frederick Merk (1963)

"The Mexican War" by Karl Jack Bauer (1974)

When most think of Mexico schemes, Aaron Burr springs to mind and how these first filibusterer attempts went down. Has any speculative "historical" fictions been written on how today's Mexico might be different had some of these early attempts at a wholesale grab succeeded?