I'm wondering if there is a more appropriate term in English that describes a rise of something (like a nation) that has already achieved that status before, like a second rise or re-rise.
We've been known to get some mileage out of the term renaissance. ("Re-nascence", or rebirth.)
In contexts less fraught with pomp and circumstance we would probably speak of a comeback.
How about resurrection? From the OED:
…3. Revival or revitalization, esp. of a person who or thing which has fallen into inactivity, disuse, or obscurity; an instance of this.
Another similar option is rebirth:
…2. A revival or renewal of a thing; a new beginning. Also as a mass noun.
2005 Geographical Sept. 80/1 The grim heroism in defiance of Nazi and Soviet horrors and the miraculous rebirth of the modern state.
In recent usage, "reascendance" is a more common synonym for "rerise". Unlike some of the other suggestions here, it means exactly rising again, and does not imply starting over.
Yet another general reference question. SIGH!
comeback, reawakening, rebirth, rebound, recovery, rejuvenation, renaissance, renascence, renewal, restoration, resurgence, resurrection, return, revitalization, revival, triumph
Poetically, you could say such a nation
rose like a phoenix from the ashes
If you'll be using this in context, then just say "rose again". As an aside, isn't re-raise used in Poker?
"Reawakening" could be a useful word. "Resurrection" is another one.
If it doesn't have to be one word, saying that a specific object "rises again" isn't bad to say.
If it's a nation you're referring to, and you don't mind a loanword, "Risorgimento".
Thinking about how I would describe China as the former and future world leader, I would speak of the return or re-emergence of China as a world leader. I would be comfortable with someone else using reascendance but it would not be my first choice. Google says re-emergence is much more commonly used but I grant that reascendence is more precise.
Rerise is not a regular or commonly used word. Rise serves both purposes most of the time.
The sun sets, and rises the next morning. You rise from sleep. The Phoenix rises from the ashes.
Similarly, dynasties that fell have risen after decades.
Rise can thus be used, with 'again' if needed.