What are ways to extract the juice from a pomegranate?
Luckily you asked for an efficient way and not a clean way!

My brother in California has numerous mature pomegranate trees on his property and squeezes the majority of the fruit each year to use for jelly.  He uses a manual Hamilton Beach Orange juicer which he says does the best job of anything he's tried.  He usually has between 16-20 gallons of juice a year so he has definitely put it to the test.  His local restaurant supply wanted around $250 for one but he got his on ebay for $150.

To minimize the mess he does it on an old table outside and covers himself with a large garbage bag by cutting a hole in the bottom for his head and the sides for his arms.