I was just wondering if there is some sweet and apt word for the phrase "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.
Later is the first word that comes to my mind.
Henceforth and hereafter are good. How about subsequently?
Hereon, henceforth, hereforthwith, henceforthwith, hereafter. Unless it needs to be specified because of prior events, it's actually redundant to state it. Some more: forward(ly), afterwards, post-present (I cheated) and later.
What about the adverb prospectively?
What about the term immediately?