Guess I'll wade in...
I'm not exactly gay about the whole organic thing- quite the contrary, it's all pretty nonsensical to me unless you accept a lot of goofy assertions. So-called 'natural' and 'organic' sprays have killed more of my plants than aphids have- neem oil especially can kill weak plants. Man didn't get to where he is by disavowing the tools he has developed, and the fresh veggie aisle in your local grocery store isn't stocked full of delicious and cheap fresh goodness year 'round because farmers are giving their crops a bath in soda pop and listerine and soup seasonings- man has developed some extremely effective and safe chemicals that work. Among them is Sevin. It works. Use it as directed and you will still be alive and your children will not develop ADHD and antennae and a third nipple. The biggest problem with most all organic methods is that the vast majority of times it's sought after as a remedy it's too little too late- with the exception of Ladybugs, most every organic method is at best a minor annoyance to aphids and the like. If they are used proactively they can keep an infestation from occurring but once it's begun you'd be best off by bringing out the big guns.
Without preventive measures you will never win the battle. You'd be best off if, seasonally, you thoroughly sterilize all containers and if you're in a greenhouse, sterilize the greenhouse as well on a fairly regular basis. The best way to do both is with a large (1 or 2 gallon) pump sprayer filled with a 10% bleach solution. Spray down ALL surfaces in your greenhouse or your deck or wherever you are growing. Spray down ALL containers both inside and out and then spray them down with a garden hose. Use new soil for planting and repotting, and it's best to sterilize the dirt either in a microwave or oven prior to use.