Generally, yes. The details are below.
Your eligibility to contribute to an HSA is determined based on your medical coverage as of the first day of each month. You must be enrolled in an HSA-eligible plan, and not have any "other coverage." Other coverage would include both the prior health plan (if it was not HSA-eligible) and the FSA. (You also have "other coverage" if you are married and your spouse has an FSA, because that covers you. And it counts as other coverage merely by existing, even if she has spent all the funds.)
For example, suppose you quit company A on June 15, but your health coverage goes to the end of the month, and you have a 60 day grace period to use up your FSA funds. Your eligibility to contribute to an HSA would start on July 1 if there were no funds in the FSA, or would start on the first day of the month after you use up the FSA funds, or would start on Sept 1 after the expiration of the grace period, if you did not use all the funds.
Also, if you plan to remain eligible for HSA contributions through all of 2026, you can contribute the maximum for 2025 under the "last month rule" even though you were not eligible for some months, as long as you are eligible as of December 1, 2025.
