First, ideal to clean the clothes again if you've noticed anything before packing to store them again. Are the insects you've found carpet beetles (little brownish-red larvae or skins, and tiny black beetles)? Not moths? Carpet beetles are a lot more common.
Having had carpet beetles, I've found the best solutions are to pack some things in plastic (just tightly closed plastic bags) and then in boxes. This will limit what anything gets into a bit as they don't seem to move around much. I use cedar balls & blocks I've sprayed in there, as well as lavender sachets. IF I've caught a bunch of bugs, I also use diatomaceous earth in or around whatever I happen to be storing materials/clothes in. I've found this is one of a few natural (or even not-natural) options for carpet beetles (If I recall correctly, I think moth balls don't affect them like they do moths - and yes, the odor does linger for several washings having gotten used moth ball stored items in the past, but will eventually go away).
It'll work great if you did it on the bottom of your rubbermaid container, then bagged clothes and were even able to put some between your layered bags of clothing, and put some around the location you're storing the whole box as well. I often have found diatomaceous earth at places like Ace hardware, or large garden centers.
I've also taken to laundering my woolens with a good woolwash that has tea tree oil and lavender oil, which is supposed to help deter wool-loving pests. So lots of options out there.