Can you get more detail on the "Top Traffic Sources"

We are a relatively small community and every once in awhile we can get a very large spike in views. When going to the Top Traffic Sources, we will get the domain (www.google.com, t.co, etc.). Is there anyway to get more detail? Namely for search engines like for Google can we find the query the user put into Google? For LinkedIn, can we find the URL that it came from (not just the domain)?
If so, would this only be done via the Data Explorer? We have the standard plan, so we don’t have access to Data Explorer.