Pocket Casts for iPhone and iPad is a well-designed and feature-packed podcast listening app, and my personal favorite for its style and queue system.
One of its great perks is that, if you opt into Push Notifications for one or more shows, you can quickly add new episodes to your queue right from the notification.
Note: You might notice Pocket Casts hasn’t been updated in a little while. I’m usually cautious of apps in that state, but it still works well on iOS 9.3 and developer Shifty Jelly is working on a huge upgrade coming soon. It will be free to existing owners, too.