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Thanks, BillM22. Please allow me to clarify: 

• Yes, my W2 box 12 shows W code with $2,000 contribution.
  I may have confused you with the statement "My new employer contributed $2,000 to my HSA as a benefit (NOT deducted from my salary)." It's not deducted from my salary, it's just a benefit. I don't lose or gain gross income of $2,000 as salary either I have HSA or not. If I don't have HSA, then my employer does not give this $2,000.
  For this discussion, let's treat it as deducted from gross income because it should be taxed at my marginal rate when I withdraw the cash from HSA.
• TurboTax pulls this $2,000 automatically from my W2. I didn't enter it. It is NOT personal after-tax contribution.
• I am actually covered for HDHP plan in November and December. But looks like TurboTax now automatically applies the last-month rule, hence I was proposing to enter October and November instead to not trigger the last-month rule. If I do that, then TurboTax correctly adds the $800 amount as taxable income. But this seems wrong because I am actually covered in HDHP in Nov and Dec instead. This is where I need some suggestions on how to do it properly. See below screenshot on TT forms:

If I select Nov and Dec HDHP coverage, then TT applies last-month rule and disregard my $800 excess withdrawal:

If I select Oct and Nov HDHP coverage, then TT applies the right taxable amount based on the $1,200 limit:

Thank you.