Not really. I think the "lockdown" was highly politicized. I don't think there was much of a lockdown anywhere in the country. Ultimately, it seems like people have been moderating their own behavior all over the country independent of official policy and we just got tired of it lately. You can see the rates of spread per state here:
Rt is the number of people each person infects. You can see the initial drop as we all had an "oh shit" moment around April. The lockdowns happened around then too, but it's not clear that's really why the Rt dropped. Really, I think people changed their behavior for the most part just on their own.
One weird thing about those charts is that there doesn't seem to be much of a correlation with geographical location between the states. Why is KY so different than TN? Is that really differences in state response or is the virus just less predictable than we think?
A note about those graphs--it's the areas under the curves that matter. So we're in trouble now because of the prolonged time we've spent with an Rt > 1. Even though it's generally much smaller than it was back in March/April, the recent Rt hump has more area under it and that's driving the infections. The frustrating thing is that it looks like if we had kept up the same level of caution as we had in April, we'd probably be fine now.