You can specify date limits on searches with by_date search parameter. Here’s a link you can add to your search results page to offer your visitor the ability to restrict the search results to last day:

<?php echo '<p><a href="' . get_bloginfo('url') . '?s=' . get_search_query() . '&by_date=1d">results from the last day</a></p>'; ?>

The date range is always back from the current date and time. Possible units are hour (h), day (d), week (w), month (m) and year (y). So, to see only posts from past week, you could use by_date=7d or by_date=1w.

Using wrong letters for units or impossible date ranges will probably lead to no results at all, depending on case. If there’s a number present, but the letter makes no sense, Relevanssi will assume the unit is days.

Thanks to Charles St-Pierre for the idea.