It does not help with the pulling, but will help protect the automatic transmission. The oil cooler should be installed so the transmission oil flows through the auxiliary cooler first and then through the truck's original oil cooler. Most of the heat will be removed by the auxiliary cooler so excessive heat is not transferred to the radiator. The factory cooler will actually warm the oil during cool weather so the transmission oil will flow properly and lubricate the transmission. If you plan on driving in very cold conditions, either disconnect or cover the auxiliary cooler.
High heat shortens the oil life dramatically.
While pulling the trailer, drive in third gear rather than overdrive. This lower gear ratio will help pull the trailer and produce less heat in the transmission. The extra weight of a trailer will put a lot of stress on the truck's brakes. Check the brakes and the brake hoses to make sure everything is in good condition. Trailer brakes will be required in most parts of the country.