The last month rule does not apply in your case, because you had other coverage earlier in the year.
“You are treated as having the same HDHP coverage for the entire year as you had on the first day of the last month if you didn’t otherwise have coverage.”
There are two parts to your question. First, can you make any HSA contributions? You are disqualified from making HSA contributions if you are “covered“ by other medical coverage. Other medical coverage includes your FSA. If you are still eligible to request reimbursement from your FSA for expenses you incur after you change jobs, then you are not eligible to contribute to the HSA. However, if your FSA coverage ended when you terminated your previous employment and you can’t seek additional reimbursement for medical expenses, then you are no longer covered by the FSA and you can make contributions to the HSA under the new employer.
Second, is the question of how much can you contribute. Since you had other medical coverage previously during the year, you can’t use the last month rule. Your eligibility is determined by the number of months in which you are only covered by a qualifying HDHP. You can contribute $285 per month if you are single and $581 per month if you are covered by a family HDHP, for each month when you were covered by eligible insurance on the first of the month. For example, if you started your new job on October 15, then you can make two months’ worth of HSA contributions for 2020 (Nov and Dec).
[Corrected to state you can use the last month rule. Just beware of the testing period requirement to maintain qualifying coverage for all of 2021.]
