My recent tax form is a free version, which does not accept 1099 form, may I upgrade it and start a new tax return?

Yes you can upgrade and you won't have to start over.  How to upgrade the Online version

https://ttlc.intuit.com/community/upgrading/help/how-do-i-upgrade-or-switch-to-a-higher-version-of-turbotax-online/00/26239

What kind of 1099 did you get?  Some you can enter into the Free Edition.

OR There is also The Turbo Tax Free File program which is free for federal and state.  And has more forms than the Federal Free Edition, like Schedules A, B, C , D, E, F, EIC, H, K-1, SE,  etc.

To qualify for the Free File website  you just need to meet one of these 3 things

AGI $39,000 or less

Active duty military with AGI of $72,000 or less

OR qualify for EIC (earned income credit.)

In order to use the Free File program, you have to start it at a special website.  Click on Sign in at the top right.  It may let you continue your return so you don't have to start over.

https://freefile.intuit.com/