I was with the employer 1 from Jan - May where I enrolled for Healthcare FSA for $500. I had some $250 rolled over from last year FSA as well. During the employer, I hardly used the FSA around $50. Because of the immigration issues, I was moved out of the employer1(loss of employment), even I was out of the country for 2 months. My FSA contributions are also not accessible after the moving out from the employer, and last around $700. I joined the employer 2 in August, the new employer is providing good HSA options.
Is it ok to enroll HSA with the new employer since there is no monthly overlapping on FSA & HSA. Going back to my original question, is it ok to have FSA & HSA in the same calendar year with the different employer in the event of employment loss?
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You are eligible to contribute to an HSA for any period that you are covered by a qualifying HDHP and not covered by "other coverage". The FSA is "other coverage", but assuming you can't request reimbursement from it now that you ended your employment, then you don't have "other coverage" as of August and can contribute to an HSA.
What I don't know is if the "last month" rule applies in your situation. You can definitely contribute 5/12ths of your annual maximum for the last 5 months of the year. The last month rule might allow you to contribute your entire annual maximum, but I don't know if it applies. @dmertz will know the answer to that one.
Incidentally, your annual maximum for 2018 is $3450 if you are covered by a single HDHP, or $6900 if you are covered by a family HDHP, plus an additional $1000 catch up amount if you are 55 or over.
A full-year's-worth of contribution can be made under the last-month rule by anyone who is an HSA-eligible individual on December 1 of the year for which the contribution is being made. Keep in mind, though, if you make a contribution under the last-month rule and you fail to remain an HSA-eligible individual throughout the testing period, all of 2019 in this case, the amount contributed under the last-month rule that exceeds the amount that you were eligible to contribute were it not for the last-month rule is subject to a 10% additional tax. The amount subject to the 10% additional tax is not an excess contribution, so it is not permitted to be distributed as a return of excess contribution to avoid the 10% additional tax.