It's OK to have persistent queues, and it's OK if messages are hanging around in the queues: Clients might be disconnected because of updates, network problems etc. That's one benefit of queues to decouple sender from receiver, and the queue is the buffer. However these use cases are not the normal mode of operation, it's rather an exceptional situation.
Using a messaging broker as "another persistence layer" is technically speaking possible, but in this case a database is probably more suitable, because quick message delivery/messaging and long term storage/database are different tools/scenarios. So ask yourself the question: Is it still messaging or is it already a database?
If in your use case the normal message delay (= period between sending and reception) is always beyond an hour, a database might be better, because JMS selectors are normally slower and less comfortable than database queries using where clauses.
There is another aspect: Consider the need for an online backup of your messages in a JMS provider, especially in a HA cluster mode. It might be easier to do this using a database.