When I have opened a blank email preparing to send and
click on the address icon next to To: I want to be able
to view the contacts by company name.

I have company name entered into the Full Name field, the
Company field, and the E-mail Display As field. Still, if
there is a person's name in the last name and first name
fields, the address book displays by that person's name
and not by company name.

To also put the company name into the last name and first
name fields corrects the problem. However, where am I then
supposed to put the person's name?

There is an article, #293217, in the knowledge base with
somewhat this same scenario, and it basically says that
the answer is to use the first and last name fields.
Again, where is the person's name to go?