One word for "any time after this moment"
Henceforth is a single word meaning "any time after this moment". Exactly which word to use can depend on context, though.
I would consider using hereafter to fill your needs. It is often used in a legal sense but can equally be helpful in other contexts.
What about simple, plain future?
Are you looking for "henceforth?" Although it includes the present moment, it indicates inclusion of all moments in the future, including this one.