Since I don't think we care about carbon pollution in a zombie apocalypse, I'm going to assume "efficient" means...
- Won't attract zombies or looters.
- Maintaining and fueling doesn't put you in danger.
- Doesn't require a lot of specialized skills.
- Doesn't require a lot of hard to find parts.
- Fuel is readily available and won't run out in your lifetime.
You're not getting any deliveries any time soon, so anything that requires a non-local fuel is out. If you have any petroleum fuel you probably want to save it to run a diesel vehicle (gasoline goes bad after a few months). That leaves wind, hydro, solar, and biomass... and maybe propane.
That means burning stuff, probably to make steam to turn a turbine. While this is relatively easy, it also produces a lot of smoke, and requires a lot of foraging. The smoke can attract looters and the foraging requires leaving the safety of your fort. And eventually you'll run out of stuff to burn. Also in colder climates your electricity generation and heating will draw from the same fuel supply.
And hey, it's something to do with all those zombies you've been killing at your fence line. You'll probably burn the corpses to keep down the smell, why not get some power out of it? A human body is hard to get started, but once the fat melts it can go for hours. How well they'll burn depends on how much fat the zombies have left on them. Eventually you'll thin out the local zombie population so they're not a viable fuel source. How fast this happens depends on what the population density was when you set up your fort, and how many your fort attracts. If your plan is to attract zombies as a fuel source... I'll move on to the next fort.
If nothing else is available, biomass will work, but upgrade quick.
If you have a river nearby, great! Even better if there's a convenient waterfall. Even better if there's already a convenient hydroelectric dam! Not too big though, that would probably be too complicated to run and maintain. It's quiet. It's unending. Incorporate it into your fort and you have power and fresh water!
The technology isn't complicated, better technology will mean more efficient power extraction. Even if you can't manage an electric generator, the raw mechanical power can help with all sorts of tasks. You don't even need a waterfall, you can put a turbine in a river and draw power from the current.
Quiet. Hidden. Unending. It's perfect... but eventually the bearings will give out. Maybe 10-20 years. Hopefully you can find some spares in that time, or re-establish civilization.
Unlike hydro, which requires a convenient river, solar will always work. In the zombie apocalypse you'll probably be relying on some sort of solar power if nothing else to keep your GPS and radio batteries charged. If you can find some industrial solar panels and batteries, more and more prevalent, you have a nearly maintenance free source of DC power. Power generation will depend on the region and season, but why not?
Ready to go solar setups can be looted from most camping stores. Everything from a small panel to keep your phone charged to a 1200Wh 12V AC power supply able to run a fridge.
Quiet. Unending. Ubiquitous. Nearly maintenance free. The only downside is a rooftop full of clean solar panels might tip off looters.
Basically the same as solar, and a great compliment. When it's not sunny it's often windy. Home installation is a bit less prevalent so scrounging up wind turbines won't be quite as easy.
The downside is the turbine, even a small one, will be noticeable to looters. And they do make some noise that might attract zombies and looters from maybe 100 meters away. Also, like hydro, the bearings will eventually fail.
Propane (honorable mention)
Wait wait wait, I thought we ruled out non-renewables? Well... propane is a special case. There's a lot of it around, and there's a lot of home appliances which will run on propane including the most important, most overlooked of all... a refrigerator! And they're quiet.
Solar and wind are unreliable, and refrigeration uses a lot of power. If your batteries run out on a windless summer night and all your food goes bad, that sucks. Having a propane powered refrigerator can save you from starvation.
Look for them in camping stores and in RVs.
So there you have it. Your best bet is hydro, if available. And you should supplement it with solar and wind. Keep biomass and propane available as a last resort.
Hoard good bearings! Efficient power generation requires something to rotate very fast. That means you need good bearings. Bearings wear out, how fast depends on how well maintained they are and how balanced the load on them is. Cheap, poorly balanced bearings will wear out in a few years. Good, well balanced, well maintained bearings might last a few decades. Good bearings require sophisticated metallurgy, you're not likely to be able to make them yourself.
They'll corrode if left on a shelf, so find them early and store them in light oil to prevent corrosion.