Over Easter weekend, when I noted the lines at the Eiffel Tower were especially long, I asked the Concierge if I could purchase tickets through him or online and he indicated that they use to be able to do this but not any more. I then checked online and noted the Eiffel Tower site allowing this. I did not proceed with online purchase as the printer in the hallway wasn't working so wouldn't have a confirmation. Anyway, not having this option was a huge bummer as I ended up on a long line.

SCflier has provided some great information here which I wasn't aware of. I can say that if you were to obtain tickets whether through concierge or online, there is indeed a line for ticket holders and it bypasses the long lines of people waiting to purchase their tickets. If you can only get tickets to the 2nd level, I did see a ticket counter on the 2nd level to purchase a ticket all the way to the top and the line there wasn't much to speak of.