Well, if your iPhone is only checking, say, one other account, you could choose that account's Inbox as your default view.
Or, you could do what I've recently done (noted by Massie, above) if you have multiple accounts and use webmail. I have a Yahoo, Gmail and the ubiquitous iCloud email account, Yahoo being my primary email account. I installed and set up their own app (a better experience all-in-all, too. But I digress) and removed it's Mail sync from iOS Mail, leaving the Gmail and iCloud accounts on the iOS's. When I see a notification badge on my Yahoo, it has priority to those on iOS Mail.
Similarly, you could do the same with a Gmail or Outlook or other mail app, I suppose - or reverse everything and have priority mail come in on iOS Mail and everything else directed to the 3rd Party app.