Okay...the T/H switch is a computer switch that lengthens the shift points, so that you will rev higher in each gear before shifting. It doesn't keep you out of overdrive per se.
Most owners seem to think that it makes most sense to use the T/H when your tow load is perhaps 1/2 to 3/4 the tow capacity of the truck---so you don't have to use T/H every time you hitch up a small trailer.
I'd say that T/H would always be a good idea if you are hauling up in the mountains, and I'd even suggest that you shift manually down to 3rd gear, even if you were in T/H, just to keep the heat down in the transmission. But I wouldn't manually downshift to 3 unless you felt the truck constantly trying to downshift/upshift (called "hunting").