No you can't efile a more complex return for free from online. Well actually all the online versions for federal and state are free to efile. The fees are the same if you efile or print and mail. You pay to prepare each return. You have to pay for state separately. In the Desktop you do get 1 state program download for free.
So in Online Deluxe and higher the state return is a separate $45.
Even though the Free Edition has a higher income limit than the government Free File website it is very limited and does not have any other schedules or forms. See this, Why do I have to upgrade from the Free Edition?
https://ttlc.intuit.com/community/upgrading/help/why-do-i-have-to-upgrade-from-free-edition/00/26379
