It's always been a thing. It's, for instance, the sense used in the 1967 Outer Space Treaty.

“Outer space” is basically everything outside of some fuzzy demarcation line (often, specifically, the Kármán line) marking the “edge” of Earth's atmosphere.

"Space" is just a shortening of "outer space" in this case. This is opposed to the "inner space", namely the space within Earth's atmosphere. Everything outside of that is "outer space".

Outer space is synonym for space.

Deep space refers to space far beyond the earth-moon system and generally earths gravity.

Interplanetary space is the space withing solar system. interstellar space is space between stars.

It's a slightly older form of the term. "Space" is just space, it's everywhere. Your car has space inside it, so does your backpack, so does your living room, it's all just ordinary space. "Outer" space has specifically referred to the space above Earth's atmosphere, to differentiate it from more Earthly kinds of spaces. Over time outer space has become less exotic and so it tends to just be called "space" colloquially these days, with the meaning understood based on context.

The term you're looking for is deep space.

You would prefer simply to say "in space" perhaps? Drop the "outer" part?

What else to say instead?

It's just a colloquial way of referring to it in a non-science context.