There just weren't a lot of such people in the US in the 1800's, at least until the Mexican/American war. No census bothered to count them, which is a pretty good indication right there. At the beginning of the century the census just counted "free white", "slave", and "free colored". After the Civil war, they dropped slave, and split non-white into "black" and "mulatto" for a while, then "mulatto" was dropped and "indian" was added, then at the end of the century "Chinese" was added.
That gives you a pretty good idea of how the 19th century mind looked at the composistion of the USA.
In the 70's where I grew up in Oklahoma it was common to refer to "Mexicans". Some would say this was reasonable due to that country's preponderance in what we call the "Hispanic" demographic today, while others considered it a kind of ugly ignorance. The later attitude won out, and referring to groups of people as "Mexicans" is now considered offensive by a lot of people. I still today get in trouble sometimes for referring to "Mexicans" on my soccer team, even though the people I am referring to are in fact Mexican citizens.
However, this clearly goes further back. The first attempt by the census to count this demographic back in 1930 was in fact a new "race" category called "Mexican". The 1890's are just as close to 1930 as 1970 is, and they most likely didn't invent that "race" name on the spot. So I think its fair to extrapolate that in the later 1800's this was probably already going on, and in fact "Mexican" was being used for what today we would call "Hispanic".