The other answers ("This is basically self-contradictory!") are good. However, I wanted to add something else you can try: you can apply a gaussian blur to the image after downsizing but before saving as JPEG. If you want to get fancy, use an L* a* b* colorspace and just blur the a and b channels, leaving L alone.
Although it's true that JPEG can produce blocky, posterized artifacts at strong compression levels, generally it has to "work hardest" when there's a lot of fine detail, and especially when that detail has strong contrast between color channels. Take a picture of bright red fall maple leaves against a strong blue sky and I guarantee you'll find artifacts even with surprisingly high-quality JPEG settings.
So, by blurring, you can reduce these details, and therefore reduce file size. This obviously degrades image quality, but you may find the result to be preferable to what you're doing now.