I'd recommend setting up new listening rules to monitor key terms and hashtags around the event or crisis. You can backfill the rule for up to 28 days but keep in mind this can take a few hours to populate.
I'd then pull these rules into Social Buzz and use one of the real time streaming options to closely monitor the conversation as it happens, and view the trend graph to monitor volume of mentions over time. Consider setting up a scheduled report if you would like to receive this data at regular intervals in your email inbox in excel form.
You can also pull these rules into Unified Moderation, and turn on streaming so the new posts come through in real time, where you can start taking action on the posts such as reply, retweet, escalate, mark as read.