Disable LTE and let it idle on 3G. Not much else you can do about it. They don't promise coverage everywhere, at least they're being upfront with you about your particular location.
You can find used Network Extenders on eBay for <$100. That's what I did to deal with the black hole (no carrier has coverage) called my hometown. It sucks, but we're in the middle of a valley shaped like a soup bowl, alongside about a dozen other households, so it really wouldn't pay for any carrier to erect a tower just to cover us.
Also, check out the BetterBatteryStats app, specifically use it to look at partial wakelocks. It's entirely possible that you have apps running in the background that are killing your battery. I found a few that were hurting me, most I could keep by tweaking settings, but I uninstalled a few that couldn't be fixed.