Die Bedeutung "auf eine andere Art" wird u.a. belegt durch Bedeutung Nr. 1 in YourDictionary:
Auszug aus http://www.yourdictionary.com/otherwise
1. in another manner; differently to believe otherwise
2. in all other points or respects an otherwise intelligent person
3. in other circumstances
4. if not: used as a conjunctive adverb do it right; otherwise, you'll have to do it over
In another way
differently, in a different way, contrarily, in an opposed way, under other conditions, in different circumstances, on the other hand, in other respects, in other ways; see also oppositely.
Antonyms like*, so*, in like manner.
Introducing an alternative threat
unless you do, with this exception, except on these conditions, barring this, in any other circumstances, except that, without this, unless … then, other than; see also unless.