Conservatism is not populism. Most "right wing" individuals, movements, and tendencies are anti-populist (anti-democratic) and aristocratic. Thus it is hard to understand anti-aristocratic conservatism.
But Jefferson was hardly an oppressed peasant himself. While Hamilton favored a manufacturing/moneyed aristocracy (an aristocracy which theoretically anyone could work his way into, Hamilton's own life being an example) Jefferson was the spokesman of an aristocracy of a very different kind: landed, semi-feudal and almost monarchical. This was an aristocracy one almost had to be born into. In Europe the Right wing is very "Jeffersonian," in that it is anti-capitalist, agrarian, and monarchic. How on earth did Jeffersonianism get the reputation of something popular and democratic?
Alexander Hamilton was not perfect, and neither is industrial capitalism. Unchecked, it is very corrosive not only of the natural environment but of traditional institutions as well, and contrary to what some capitalist utopians think, it doesn't reward everyone who tries to play the game (the myth being that only the lazy suffer). But still, Hamilton was far preferable to the armchair Jacobin Thomas Jefferson.
Jefferson's admirers also continually overlook his (and his party's) support of the French Revolution (and nowadays neo-Confederate Jeffersonians have the unmitigated gall to accuse Hamiltonians of being "Jacobins!") as well as his notorious religious heterodoxy and radicalism. The Jeffersonian party may look like the "party of liberty" to people safely ensconced in the twenty-first century, but at the time people were terrified that a Jeffersonian victory would lead to Jacobin terror and irreligion here as had happened in France.
There is no "official" interpretation of the original intent of the Constitution. From the very beginning there have been two schools of thought represented by Hamilton and Jefferson. But neither are implicated in the horrendous situation in which we find ourselves today.
Jeffersonian "conservatives" (particularly the neo-Confederates) need to ask themselves what business they have attacking "aristocracy" and celebrating "the people." The latter is best left to the Left.
The arch-Jeffersonian republican was John Randolph of Roanoke who famously quipped "I am an aristocrat. I love liberty. I hate equality."