Yes it's possible, you can open an HSA anywhere in addition to the one through your employer.

However, you should always contribute via payroll when possible to avoid FICA taxes. Sometimes you can convince your employer to deposit your payroll HSA contributions to the account of your choice, but most of the time you have to use theirs for that method of contributions.

In that case you will need to transfer regularly from your work HSA to your preferred one. Two ways to do this:

1. Trustee-to-trustee transfer. This is similar to transferring money from one bank account to the other. It goes directly, and the IRS places no limit on how much you do this per year. However, some HSA providers charge a fee for this, so avoid it in that case and instead do:

2. Indirect Rollover. You manually withdraw the funds (via transfers/checks) and then deposit it into your preferred HSA, and fill out some forms. This doesn't cost fees, but the IRS only lets you do it once every rolling one year period.