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You wrote "less than or equal to 5.4" in the title. I think you want a phone with a large screen.
If battery life is your priority, the Verizon-exclusive Motorola Droid Turbo 2 and the Samsung Galaxy Note 5 have the best battery life.
Selling Droid Turbo 2, refurbished. $200
Bought it on Amazon Cyber Monday but decided not to use it. See the listing below for pics and details.