Ok, but what about the initial check? Is there one? Or should the "first one" be made when you move to cover for the first time?
In other words, borrowing from your own post:
If we remove the initial "from cover" clause, the first check is made when you move to cover. You keep that result until you move to a new covered position, at which point that is now your check.
Under "routine" circumstances, the DM could allow passive Stealth to be used, perhaps only calling for a check when any passive Perceptions would automatically beat the passive Stealth.