I tried many 3rd party apps out there including Outlook, Boxer Pro, Inky, CloudMagic, and many others. My key requirements were:
1) Works with a variety of popular email services (iCloud, Gmail, Hotmail, Yahoo, etc.)
2) Ability to attach documents from iCloud Drive, Dropbox, and/or OneDrive.
3) Newsletter and other HTML emails format properly.
4) I could move an email from one account to the folder of another account (for example, move a Gmail email to an iCloud email folder).
The ONLY one which could satisfy all 4 criteria above was the stock Mail app under iOS9.
I had high hopes for Outlook but I find it still too buggy and it won't let you move emails to another account. Inbox and Gmail only do Gmail.
The other ones tend to mess up the formatting of HTML emails, especially newsletters. For example, there are certain emails that are formatted so they look nice on a phone and a desktop. I found most of the others couldn't handle this properly...they would use the desktop version or they had too much margins on the sides of the email on the phone version.
None of them (from memory) allows me to move an email from one account to another. The reason this is important is that I want to have one folder for a variety of themes, e.g. Purchases, Travel, School. I don't use the same email address for all of these, especially purchases. Sometime I buy with iCloud and another time I may purchase using Gmail (due to geographic restrictions). I'd rather have all purchase emails stored in one folder so if I need to search on them (say I'm reconciling my monthly budget), I can find in one folder.
Mac Mail under iOS 9 is fine for just about every need. The others have a lot of catching up to do.