It's almost tempting to reach for "elsewhen" or "elsewhy", since we have "elsewhere". But if saying "elsewise" feels like chatting with a very elderly relative, saying those ones feels more like a seance with distant ancestors. But apparently it used to be a thing!
https://en.wiktionary.org/wiki/elsewho#English
https://en.wiktionary.org/wiki/elsewhither#English
https://en.wiktionary.org/wiki/elsewhence#English
https://en.wiktionary.org/wiki/elsehow#English
https://en.wiktionary.org/wiki/elsewards#English
I my head as a child I instantly understood and thought "Ahhh, they mean like 'OR else this happens.'
I have always assumed it was a boolean logic thing meant to help show that it wasn't an AND, but 'or' had already been used as a keyword so they did 'or else' and dropped the reserved 'or'.