What's the correct thing to do? I can't even remember in the US... put it back in the mailbox with a sticky note or something?

I suppose I could ask at the post office. It's from the Department Gesundheit und Soziales, Abteilung Gesundheit, for my canton. And it's a 1 unit rental property (house) and I've been here almost a year. Right adress, wrong name.

Take it to the post office. They know what to do.

1) Do not open it.
2) Write "Empfänger unbekannt verzogen" (recipient moved to a unknown place) on the letter.
3) Drop it into the next yellow mail box or hand it in at the local post office.

If you have time to spare you could give the Abteilung Gesundheit a call before returning it as described above. Without opening the letter of course.

Reason being, this could be in connection with health insurance premium reduction by the Canton, and AFAIK the recipient needs to act somewhat swiftly in those cases. Just returning the letter as described by aSITUS will cause delay, possibly more than allows for a timely reaction to the next letter at the new address.

Well the first thing you do is check it is for one of the neighbors in your apartment block or in the next door building. In 30 years I can’t recall a single case where I was not able to solve the problem that way.

Otherwise do as others have suggested.