Any number can easily be moved to the top of the list. Long press the number and you get a pop up option where it can be made the default number. Once default it will be at the beginning.
For the rest of them I was able to move numbers around by copying and pasting around on the phone.
But I wonder, if you install an O.S. upgrade then re-sync with google, will the ordering be preserved?
Its not quite what you are asking for but I'll point out that the contacts/dialer app lets you display contacts only categorized to a particular group. For your purposes it might be best to create a group with the company name, then create individual contacts within that group for the different numbers. This allows for more descriptive names: one contact with 15 numbers all labeled 'work' sounds unwieldy.
The downside to this is that you might have to frequently change what group you want displayed, but that takes a couple of seconds.