No, sorry.
From IRS Publication 969:
Other health coverage.
You
(and your spouse, if you have family coverage) generally can’t have any
health coverage, other than an HDHP. However, you can still be an
eligible individual even if your spouse has non-HDHP coverage provided
you aren’t covered by that plan.
You can have additional insurance that provides benefits only for the following items.
Liabilities incurred under workers' compensation laws, tort liabilities, or liabilities related to ownership or use of property.
A specific disease or illness.
A fixed amount per day (or other period) of hospitalization.
You can also have coverage (whether provided through insurance or otherwise) for the following items.
Accidents.
Disability.
Dental care.
Vision care.
Long-term care.
