The solution is easy.
Depending on the mailbox, on either the outside, or inside (but visible when opened) put the last names of everybody who lives at your address followed by the word 'ONLY'.
If the mail is addressed to the old tenant but includes 'or current resident' the USPS must deliver that mail to your box.
Did you also know that it is against postal regulations for you to write on any mail that is not addressed to you? That would be defacing mail. The best thing is to attach a post-it note to the item.
It is also necessary to remember that the post office does not return mail to the sender just because you write that statement on the mail. 1st class mail is forwarded to the new address for a period of 1 year.
Do you want to receive mail at your new address that was sent to your old address? Of course you do. So you would not want any new tenant at your old address writing 'return to sender' on those Christmas cards would you?
Contrary to belief, the USPS does not have a master list of everybody that lives at every address in the US.
In situations like yours it is very likely that there is not a regular carrier who is servicing your route everyday. It is likely that there are multiple carriers who carry the route every week. That makes it difficult on the carrier and the patron.
I hope this helps.