iPhone 4S battery won't last even after new one installed?

I bought a used iPhone 4S this summer and everything looked in great shape except the battery drained quickly. I replaced the battery with one from iFixit and everything went smoothly, but it didn't improve battery life at all. I've followed every battery saving tip I can find, but nothing seems to help.

With bluetooth off and WiFi turned on and little to no use, it will last a maximum of 7 hours. If WiFi is turned off, it will make it 12 hours before needing charged.

I've reset/reinstalled iOS several times. I thought I would give iOS 8 a try and see if anything was better - long shot I know. Now the phone with WiFi/Bluetooth off won't make it more than 5 hours.

I don't know where to go from here - what part(s) would you recommend replacing? If it were WiFi, wouldn't the battery last much longer once it was turned off? Thanks for any suggestions!