Yes, as long as your general purpose FSA and HSA don't overlap, and (if you're married) as long as you're not eligible under your spouse's FSA.
You can't contribute to an HSA and have a general purpose FSA for overlapping months.
If you are eligible for medical expense reimbursement under a spouse's FSA, you are disqualified from contributing to an HSA.
Your new company's benefits officer should be able to answer any other questions.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
