Payroll has nothing to do with this.
it is the same as a 401k: you dont tell payroll to put your contributions with another company, you transfer it yourself.
you can transfer it manually yourself every month.
or dont let your company take your HSA contribution out of your check.
You can make the contribution yourself post tax and then get the taxes refunded back at tax return time.
but if your company just hands you HSA money for selecting a HDHP they usually give it to you as a lump sum in the beginning of the year and you can transfer that the day it hits in january.
Not quite. You CAN move your HSA account elsewhere, but the OP is correct that he can't contribute thru payroll (and save FICA) if he goes elsewhere. It also affects other things, like FAFSA if you have college students (payroll contributed to HSA is invisible to FAFSA, but added back if tax is refunded thru the 1040), and EITC (compares line 7 wages vs AGI).
OP, you need to agitate your employer to switch HSA providers, or apply leverage to reduce fees. DH's employer got a promise of no sign-up fees from their provider Key, since they have other accounts thru that bank. After a few months I was not pleased to see $3 monthly fees coming out of the account - they added a paper statement fee, and buried the notice on the back of statements. When I complained to Key, they pointed out that I could just receive e-statements for free. This is one of the biggest reasons I am not happy about the FNFG-Key merger, I like FNFG, but Key is sneaky.