In the same vein of arriving early, have you considered going somewhat near closing? Also, I know that these aren't in Rome and therefore isn't a truly accurate comparison but... last year we walked by the Anne Frank house in Amsterdam about an hour before closing and the line up was close to non-existent (though we decided to walk around the Jordaan neighbourhood instead). This year, we went to the Uffizi about an hour before closing and the wait to get in was < 10 mins. We didn't go in (we went four years ago) but the friends we were traveling with went in.
Obviously by doing this, you might be restricting your time at the attraction if they kick you out so save this strategy for the attractions on your B-List.

Watch out for line hoppers. There was a pretty wacky argument behind us where some group accused another group of line hopping and the Vatican Museum staff didn't want to get involved at all.
Bring water to stay hydrated and a hat to help stay a bit cooler while in line.