I use logicmail (that can be dl ota on this site). It does not have push. But I use the forward as txt feature of gmail to be alerted to new mail plus it sends the first several lines of Text.
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Unfortunately Gmail app is not as great as we considered it would be. Initially I thought it will support all email clients, but it only works with Gmail accounts. This is where I discovered its good alternative MailWise which provide clutter free email experience. Everything from Gmail, Exchange, Outlook, Hotmail, Yahoo etc. it unify all account at single place. It group all notifications of social network updates, events, newsletters etc. create best conversation views, strip out excessive formatting and email signatures and even provide client-side privacy thus no one can read our private emails. I think you should try it too.
