What is the correct word to use instead of else’s?

I would say, "That's an SEP". Anyone who knows me would know what I mean.

In any event else's is perfectly fine. Dictionary.com's entry for else says, "other or in addition (used in the possessive following an indefinite pronoun): someone else's money."

There is nothing wrong with "someone else's". Inexplicably, many spelling checkers don't automatically recognize possessives of user-added words, so don't try to read much out of that.

From a grammatical perspective, the key insight is that -'s is not a word clitic, it is a phrase clitic. For example, it may be informal or even a little unidiomatic, but it is not ungrammatical to utter or write sentences like the following:

I take exception to the senator from Maryland's remarks. The package that arrived on Monday's return address was somewhere in Texas. The man I was talking to's attitude really bothered me.

Worrying about the placement of the possessive clitic is someone else's hang-up.