The TurboTax online editions for tax year 2020 should be available in early December.
The TurboTax desktop CD/Download editions for tax year 2020 should be available by mid-November - https://turbotax.intuit.com/personal-taxes/cd-download/
Although the beta versions of the program come out early they are not fully functional until sometime late January at best and many forms including the state programs are not complete until late Feb or sometime in March. The IRS efile start date is still TBD however in the last couple of years they have not opened until very late in January so don't be in a big hurry.
Please be careful about being in too much of a hurry to file your next tax return. Every year we see people who are in such a rush to file that they file incorrectly and incompletely. If you file too early and leave out a W-2 or a 1099, or forget to enter information, you will end up having to amend your tax return---and that takes months. Employers have until the end of January to issue W-2's----and ALL of your 2020 W-2's have to be on the same tax return. Many of the 1099's and 1098's you need do not arrive until late January or sometime in February. If you are getting Earned Income Credit or Additional Child Tax Credit, the IRS is going to hold your refund until the end of February no matter how early you file. There seems to be a greater chance of "early" returns being reviewed and those refunds are delayed, or early returns are more often subject to identity verification--which will also delay processing. Filing too early may just be a recipe for a delayed refund. It might go more smoothly for you if you wait until at least mid-February to file instead of e-filing during the first minute that you can do it.