As long as you are legally in the United States you should be able to apply for primary, secondary, or even tertiary insurance.
Know that having a secondary insurance does not mean you are exempt from the deductible of the primary insurance. In fact you very well could have a deductible for both.
Depending on what your income level is you may qualify for income based insurance offered in some states.
If you don’t like the plan they offer you could ask your employer if they would be willing to compensate you the amount they would pay for insurance. Even a percentage is better than none. It’s worth a shot and I have seen it happen before.
Before picking a secondary insurance make sure that they cover doctors in your area.
A lot of employers offer Flexible Spending plans that might be what you’re actually looking for.
Unfortunately, there isn’t a short answer on which insurance plan to pick. Just research the market and even make some calls. Balance out what you think you might/could spend with how much you could save.