Generally no. The 1040-X itself, and the attachments & schedules TurboTax will generate with it, will adequately explain to the IRS why you're amending your return.
That is especially true this year when thanks to COVID delays, the IRS is advising everyone to e-file as much as humanly possible. (Even electronic returns are seeing delays this year, but not nearly as bad as paper returns.) Though TurboTax won't have its 1040-X ready until March 25 (just checked it a few minutes ago -- I have a situation that needs amending myself), once ready that will be an e-filed extension; the IRS just started accepting 1040-X for e-file last tax season. Allowing users to add their own cover letters to an e-filed form only makes things even more complicated for both TurboTax and the IRS.
I see you specifically added "by mail" to your question. Still, (a) given the nature of the IRS' COVID-related delays ANY paper filing with the IRS should be avoided if at all possible, (b) the 1040-X no longer has to be filed on paper as of last tax season (as it was for many years before), and (c) a cover letter may still confuse the IRS when TurboTax should automatically generate the statements, schedules, etc. to be included with a paper 1040-X.
well, the thing is I didn't use Turbotax to file my return in the first place, and the software I used doesn't support "amending electronically". So at this point, I don't know If I can amend electronically.
The Turbo Tax 1040X Amended return won't be ready until the end of March. And you can only efile an amended return if the original return was efiled and accepted in the same Turbo Tax Account.
Maybe the software you used to file will let you efile the amended return later. I don't think the IRS has it ready yet either. They only have the 2019 1040X listed.