Health Equity kinda sucks, but as others have mentioned, you can do a trustee to trustee transfer at any point.

If they charge a fee, you can instead do a yearly (rolling 365-day period, not calendar year) indirect rollover where you withdraw money like you would if you're reimbursing yourself, and then deposit into your new HSA. If they give you checks, just send a check with a form to Fidelity . Either way, make sure to tick the rollover checkbox so it doesn't get coded as a contribution. You need to complete the transfer within 60 days or it'll be considered a withdrawal instead.

I now have Optum Bank which sucks in many similar ways, and I do regular transfers and haven't had any issues for ~2 years now, so it should work fine for pretty much any HSA. Just pay attention to fees in case your HSA assesses one.

Another less likely option is to ask your payroll department if you can just have your contributions to straight to your new HSA. Some employers allow it, some don't.