Well...
#1 - You're trying to book a transatlantic fare at the last minute at the height of tourist season.
#2 - ITA doesn't automatically look for the fare with the most segments. You have to force it to.
#3 - You haven't specified an airline you want mileage with, so we don't know where to even start to look for you.
Poking around a bit, you may be able to do LAX-MEM-DTW-CDG on NW. Or maybe try routing through FRA on UA...that'll give you a few extra miles. But I wouldn't be expecting anything cheaper than what you've already found.