What do you mean by "provided"? Did your company pay for it?

Booking class T is a discount revenue booking code. On international itineraries it is not upgradeable other than doing an upgrade for cash at the airport (if offered...would cost anywhere from $600-$1,000 depending on the route).

T does earn miles and you would be able to upgrade the domestic segments using e-500's or other upgrade instruments.

The only caveat to all this is... if this was some kind of free ticket that UA gave to your company it may have mileage earning restrictions. But since it's T you should be able to upgrade the domestic segments.