There is no process for you to upgrade from the Free File Program to the TurboTax online editions.
You will need to start a new tax return using the online editions with a new User ID - https://turbotax.intuit.com/personal-taxes/online/
Turbo tax may have fixed this because I was not finding a way to switch to a paid version after starting the federal free file product. I clicked your link and answered the boxes about my tax situation and it recommended Turbo Tax Premiere. I logged in, still throufh the link and it asked if I wanted to switch from gederal free to Premiere. I confirmed and my information was all transferred over to the Premiere product. I am glad it did as it was a couple hours worth of work.