How to get out of the queue after three minutes to get user input?

You can combine the default sample queued callback flow https://docs.aws.amazon.com/connect/latest/adminguide/sample-interruptible-queue.html with the Voicemail solution https://aws.amazon.com/solutions/implementations/voicemail-for-amazon-connect/

the trick here is that the voicemail solution is user base, so what you need to do is create a "queue" user (like if you have a queue call "BasicQueue", create a user name "BasicQueueVM") if you want to have a group voicemail. So the portal will take this user in, you can then give it an extension, most importantly, the email address for the user should be a group email to all the agents in the queue so they can grab it.

Subsequently, you use Task base voicemail from this blog https://aws.amazon.com/blogs/publicsector/how-to-create-task-generating-voicemail-solution-amazon-connect/ with the same technique of a 30 sec timeout in the Loop prompt to ask if they want to leave a voicemail. But be aware that the Loop prompts will start from beginning again instead of resuming where it left off at 30 sec if caller decides to continue waiting.