The '06 Silverado has a 4 spd transmission and doesn't have the tow-haul programming of the newer transmissions. So you have to pay a little more attention -- we had an '05 Suburban, for what its worth. Often, or maybe usually, I would just leave it in "3" for most of the trip, since the engine didn't have power to tow in 4th gear anyway.
Since your trailer is lighter and more aerodynamic than my 6500# box, you might be able to hold 4th gear, especially if you have 4.10 gears. But if it is 'hunting' a lot (ie, shifting between 3 and 4 often), you may be better off just leaving it in 3rd.
Other than that, make sure to use the gears coming down the hills. Downshift as necessary to control your speed. We often shifted down to 2nd, and even 1st on tight windy roads. The last thing you want to do is to be on the brakes too much!
I always try to look ahead and use the yellow recommended corner speed signs. They may be a little slow when not towing, but find them to be usually spot on when I'm towing.