I was banned from recommending non-organics by Waira himself. And for good reasons, but that's in the infirmary.
NEEM works well on contact, and when the bugs eat the tissue that has been sprayed. It is best as a preventative.
Permethrin... that is the heavy duty stuff. I use it every year, non toxic for humans, dogs, and most mammals, but IS toxic to cats or fish for some reason. It always keep a bottle around, I used to get rid of peach tree borers last year. I also spray it on new construction projects. It will live in the wood and prevent termites for up to 10 years or so. It does NOT persist in soil, it gets broken down by soil bacteria.