I am a USPS carrier, and I run into this on the daily. There are several variations of this solution, but it ultimately comes down to the former tenant needs to put in a change of address form. If they're a postal worker, they should've done it (it takes literally less than 5 minutes). They may be trying to dodge debtors, as mentioned earlier. I would definitelt put a sticker on the box saying "(new tenant's name) only," because chances are there are carriers who rotate that route daily, and it's hard to keep up with all the forwards, especially on multi-family addresses. Another thing that will help is instead of writing "return to sender" on the mail, smply write "FWD." This will force the mail back into the system, and if there is a forwarding address on file, will send it on to where it should go (assuming it's first or second class mail). If there is a forwarding address on file, it will expire after 1 year, at which point you may have to start the proccess again. Another tip, which you may or may not want to bother with: if the mail says "first-class" or any version of "service requested" ie change service, it can be forwarded. If it simply says "standard," it cannot be forwarded, and most likely will end up in the recycling bin. But the safest bet is to rubber-band the whole bundle, mark it FWD, and leave it for the carrier. As for going to the post office to rectify the situation, I would request to speak to the actual Postmaster or a Supervisor, as they will have the most accurate information, as well as the most influence over the situation.
I know this is probably an overly long explanation, but I hope it helps.