A combinator would require it (e.g. a combinator has to report how much of the input it consumed in the process of producing its synthesized result). I should add it.
The combinator calls the secondary output "remainder". Transcode calls it "next", but then variables tend to put it in something called rest.
The idea I had was that if the signature on a function had something called input and then this remainder/rest then UPARSE would be willing to call it...filling in input from the current position, and updating the position based on the remainder.
Here's a semi-related note on the subject of copying and INTO