VPN Access

Sometimes I cannot browse to sites that are inside my corporate VPN and I get an error from Cox saying the site cannot be found.  An example might be http://myHR, which is not a fully qualified domain name only accessible from inside my company's VPN.  Sometimes it goes through and sometimes it appears that my Cox internet connection does not understand that this is a VPN URL and not an internet site.

My question is, is there a way to set up my home network so that my service will know to pass this URL along to my VPN connection and not try to go find it out on the internet and thus produce the Not Found error.