I was recently working with someone on the definition of High Deductible Health Plan. For purposes of being eligible to contribute to HSA, the health coverage has to specifically meet the IRS definition of HDHP. The people I was working with have a plan that, as an example, they pay the first $250 out of pocket, then they have a shared out of pocket, up to some amount like $2,500, then the plan kicks in at 100%. They considered that High Deductible, and opened an HSA. That isn't a HDHP as defined for purposes of HSA.

The employer or the employer's insurance broker can tell them if their coverage qualifies for having an HSA, even if the employer doesn't offer participation through payroll.

"For calendar year 2024, a “high deductible health plan” is defined under § 223(c)(2)(A) as a health plan with an annual deductible that is not less than $1,600 for self-only coverage or $3,200 for family coverage, and for which the annual out-of-pocket expenses (deductibles, co-payments, and other amounts, but not premiums) do not exceed $8,050 for self-only coverage or $16,100 for family coverage."