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"Redouble" is almost always used in the idiom to redouble one's effort, meaning to increase the effort one is exerting. "Double again" has the very specific meaning of This was increased by 100% ...
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#3: Attribution notice added
Source: https://writers.stackexchange.com/a/20749 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
"Redouble" is almost always used in the idiom _to redouble one's effort_, meaning to increase the effort one is exerting. "Double again" has the very specific meaning of _This was increased by 100% of the original, to make 200%, and will now be increased by 100% of that, making it 400% of the original._ Which one you want to use is dependent on context and your intended meaning.