The thing is that an "employer contribution" refers to how the contribution is handled (i.e., coming from the employer to the IRS), and not the source of the funds. This causes no end of confusion, as you can imagine.
Contributions "through your employer" are assumed to have been removed from Wages in boxes 1, 3, and 5 on your W-2. But you do not have a W-2 for this amount. Therefore, it can't be treated as a contribution through your employer. Every other possible contribution to your HSA is treated as a "personal" contribution.
The "employer contribution" is tax-advantages because it is removed from Wages. Every other HSA contribution ends up on line 13 of Schedule 1 (8889), because there are no Wages to remove it from.
So, unless your ex-employer removed this $656 from Wages on your W-2 (the one that they did not give you), reporting it as a personal contribution is the right thing to do.
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