Just thought the same myself, glad the search feature here works 😛
Often I just want to follow up with more explanation rather than start from scratch. Honestly I'd say that's the majority of the time. It could be two buttons "Continue in Assistant" which maintains context and just lets you continue with further questions, or "Ask Assistant" which refreshes the question (could be top of the page).
I like the Bing method as well, with an in-line question box that just would jump to assistant when questioned and full context.