Yep, I believe enabling the combined log format in Apache will give more detail–though I’m not sure if it will give all of the detail you’re after. It adds referer and user agent. Referer, I believe will result in the search terms showing up.
To enable this, you’ll go to the Webmin Apache module, edit each domain that you’d like to provide the additional information, and click the Log Files icon. On that page, find the “Access log files” option and change the format from “common” to “combined”.
If you have a lot of domains you want to change, you can use sed:
cd /etc/httpd/conf
sed -i "s/^CustomLog(.*)common$/CustomLog1combined/" httpd.conf
If you need more data than that, you’ll have to add another custom format. The Apache docs cover logging nicely: