I know this isn't what people want to hear (that's what I would say), but Google's GMail app has become really fantastic. I noticed it was a 4.5 star app with 5k+ reviews, and when I first tried it MANY months ago it was around 2.5 stars. So I figured I would give it a shot (I currently do have the benefit of "grandfathered" Exchange though). I ran this side by side with my mail app and the gmail app actually gets the notification a split second sooner. Because of how iOS works it doesn't actually download any part of the message before you launch the app though, so that might be a problem for some.
For those who have not recently tried the GMail app, I highly suggest you give it a go. I was a pessimist too. Of course, if you have multiple emails, this doesn't exactly help you much in terms of convenience.