It used to be called paleoconservatism, but that term has gone out of favor. As Milo Y. points out, there are some new elements in the movement, such as a segment doing/saying outrageous things for the lulz, and then the disgusting 1488ers. There was a small but outspoken group of antisemites in the old paleocon movement, and unfortunately they are still around.
Paleoconservatives and Traditionalists are not alt-right any more than Gary Johnson is an actual libertarian. The alternative right is racialists and a mirror of cultural marxism. Paleo-conservatives were a traditionalist response to neoconservatives. But it is more than just a generational thing. Richard Spencer, Richard Hoste and many intellectuals of the alternative right oppose Classical Liberalism and Classical Conservatism. They are neo-reactionary romantics who look to proto-fascists like Evola, Corneliu Zelea Codreanu, and modern fascists like Aleksandr Dugin. These people are completely outside of the standard American political spectrum. They are not paleoconservatives and despise Russel Kirk.