I use Podcast Republic (on Android though, dunno if its available on the iPhone). Its excellent, nicely laid out, very straightforward to use and doesn't automatically delete anything. You can set up subscriptions which take you straight there to download once a new episode is ready. Definitely the best I've used (and I listen to loads).
I'm on the public beta of iOS9 and it looks like Apple have made a right pigs ear of the standard Podcast app (which I used for years without any problems). It's possible of course that the final version that comes as part of the full iOS9 launch will be much better but it's good to hear some other recommendations ...