In some cases, yes. Please order a QuickSwitch on your new shared hosting account [within 60 days] through https://cp.westhost.com/ to have Support check your account for the possibility. Free QuickSwitches are only available for shared cPanel transfers.
If your account does not qualify for a QuickSwitch you can also contact our web design team to see if we can move your site(s) at an hourly rate. Submit a free quote request here:
http://www.westhost.com/website-design/design-quote/
No worries, Our experts are here to help.