Pests attack pests most when the plants are stressed, so it is not surprising that the spider mites appear during the most plant stressful time of year for you. The first thing you can do to reduce the pests is to reduce the stress on the plants, so you may want to increase water to the plants during this time.
Second, they will need multiple treatments and you may want to start treating from the start of the season rather than when the pests start to be seen. This will prevent the early infestations and reduce the problem over the course of the whole season.
You will need to treat several times, especially while they pests are most active. You can kind of think about it as dealing with waves of the pests, as new ones will be hatching all the time. But, the more adults you can kill, the less they can lay eggs, which means fewer adults you have to deal with in the next wave.
Third, consider seeding your garden with ladybugs or predatory mites. With plenty to eat, they will be happy to stay. If you use neem oil in combination with predatory bugs, the neem will kill the spider mites but won't hurt the predatory bugs.
I think the combination of these will help you significantly. There is not one treatment that I can give you that would work any better than what you have already tried,