Hi,
I believe the easiest solution is to make two separate search bars, one for the Future events and one for the Archived events. Then create a filter for both cases, so that events before/after are filtered out. If the frontend search settings are disabled, these filters will be hidden from the user.
Future events search bar
Create a filter like this: https://i.imgur.com/rdWVWib.png
Archived Events search bar
Filter: https://i.imgur.com/cBqVS7F.png
In these examples I’m assuming the dates are stored in a datetime format. It works for most cases. If you know it’s stored differently, then change the Date storage format option accordingly.