I just checked their style guide (it’s an Autistic fascination of mine 😅)… funnily enough, they specifically recommend using “far-right” over “alt-right”. But they don’t have an entry for “far-right”, and I couldn’t find any mention of “hard right”.

“alt-right”
use once and in quotes at first mention; use far right at any subsequent mentions. Note: far right (noun) and far-right (adjective). More specific descriptions can be used where relevant when mentioning individuals associated with the movement, but such descriptions should be evidence-based.