I'm afraid there is no built-in way to do this, redirect vars only work for form and with fixed values.
One way around it would be to write your custom variable to the user session storage (https://docs.joomla.org/How_to_use_user_state_variables )
in the first search using one of the events (before render should work), then get this variable in the second search using the same event and getUserStateFromRequest and set JInput variable with the same name using that value, this way seblod would think this variable is set in the url.
Please not this is just an idea, it migh be it won't work due to the execution order e.g. if seblod grabs its input variables before event is executed.