Can I purchase Supplemental Health Insurance while covered by a HDHP and HSA?

Can I purchase supplemental and/or secondary health insurance while being covered by a HDHP and contributing to an HSA? I was thinking about purchasing a secondary and/or supplemental health insurance plan so I wouldn't have to dip into my invested HSA funds if I became sick, hospitalized, or was injured. I was thinking about purchasing some kind of Gap policy, an accidental medical expense policy, or some other type of supplemental health insurance plan to help cover my deductible. Can I do this while enrolled in a high deductible health plan and maxing out an HSA?