General:
* Alternatively
* On the other hand
* Conversely
* In another case
* If not
* Or else
* Else
* Instead
* Rather
* Differently
* Contrariwise
More specific:
* For another reason: If not, except, but, save, barring
* To the contrary: On the contrary, to the contrary
* In a different way: Differently, dissimilarly, incongruously
The best synonym will depend on the specific context of your sentence.
For example:
* "I'm going to the store, otherwise I won't have any milk." (You could replace "otherwise" with "or else" or "if not")
* "The movie was good, otherwise it would have been boring." (You could replace "otherwise" with "on the other hand" or "but")
* "The cat is otherwise healthy." (You could replace "otherwise" with "except" or "save")