If you're needing it to use different default routes based on which local IP address is being used, this is where iproute2 comes into play.
A "name server timeout" indicates that the name "mycompany.com" can not resolve.
What happens if you do a ping mycompany.com (NOTE: Because of MX records, a failed ping does not mean it's impossible for mail to go there)
While not quite the same as doing a ping text, what does host mycompany.com give you?
Some other useful information can be gleamed from ...