Curious why you didn't keep your current Volusion URLs? With BigCommerce (which we are a partner), you can match the Volusion URL and forego the 301 redirect if you feel that's critical to your SEO success. A proper 301 redirect should be adequate and not account for such a dramatic drop of traffic.
As an ecommerce consultant you need to understand where your traffic was coming from and make sure that's protected when you move. For example, if your products have a lot of Facebook Likes, when you change the URL, you will lose 100% of those Likes...and this should be a factor in choosing a new platform.
You should be able to diagnose what's happening from your Google Analytics. Take a look at where traffic has dropped. What landing pages were driving traffic before and no longer are? What keywords have disappeared and what pages were appearing for those keywords? Are those pages 301'd correctly? What products and categories were driving your traffic and revenue?
Obviously check all your pages for proper meta tags (INDEX, FOLLOW)...with these "simple" platforms, it is easy to accidentally make your whole store disappear.
It sounds to me like something was missed during the migration and you're now. We are a top Volusion consulting firm and have worked on hundreds of stores...and we're partners of both BigCommerce and 3dcart. You can have a successful store on a hosted platform. Having PHP access is not critical to your success. Yes, it can help but you migrated from Volusion which has much more restrictions than BigCommerce (in terms of what you can accomplish from a template perspective and SEO)...so I feel we're missing something here.