You could update your storage as you go, the indexing doesn't appear to be that expensive.
For many tasks it wouldn't be helpful because the input is small enough to be covered by the context already, and for summarizing and question answering tasks, you want it to repeat information from other documents, but not from earlier in its own output.
It might be interesting for a long-context task like "given the first parts of this book, complete the next chapter".
