This prior question is relevant: A Latin adjective for New York?
You could make a compound adjective Novohispaniensis/hispanus/hispanicus (compare Novocomensis). "Novohispano" has been used in Spanish, which if borrowed into Latin would give us Novohispanus. In this context, the terminal -o- of Novo- is presumably the linking vowel -ŏ-, borrowed from Greek.
Here's an example of "Novohispaniensem picturam" from Google Books, although I'm not sure of the context.
Another option would be Neo-Hispaniensis/Hispanus/Hispanicus (with neo- from Greek νεο-).
"Novihispanicus" has been used as a species name, but biological Latin doesn't count for much.
Novae Hispaniae, the genitive of Nova Hispania, is certainly useable in many contexts.
The difference between Hispanus and Hispaniensis is said to be that Hispanus refers to a native whereas Hispaniensis refers to a non-native inhabitant of Spain. It's not clear to me how this distinction would transfer to the context of New Spain.