The black Stetson was actually worn by air cavalry troopers in Vietnam, and still worn in cavalry units today. The 1st Cavalry Division Association credits the practice to LTC John B. Stockton, in 1964. Stockton commanded 3rd Squadron, 17th Cavalry, part of the testbed 11th Air Assault Division. When the 11th was redesignated 1st Cavalry Division in 1965, 3/17 Cav became 1st Squadron, 9th Cavalry, Kilgore's unit, and deployed to Vietnam with the division. By 1970, when the movie takes place, the black Stetson would have been a common sight in cavalry units, with the exception of D-3/5 Cav, who wore the grey slouch hat. Here is a photo of Stockton wearing hat, scarf, and spurs. It was an esprit-de-corps thing, separating the Cavalry from mere infantry. The scarf can be worn with the hat as part of the uniform.

The Air Cav guys were a different breed, with some colorful individuals. According to Rex Gooch, in his biography of Alan "Ace" Cozollio, then-1LT Cozollio spotted an enemy soldier dive into a rice paddy. Cozollio then proceeded to land his OH-6, pull out the heavy cavalry saber he kept with him, and start probing the murky water with the saber until he found the man, who he captured and brought back. Ace and some of the other officers had 1860s uniforms made.

So, as commander of 1/9 Cav, Kilgore would absolutely have owned a black Stetson, and likely the scarf. It wouldn't be improbable that he owned spurs as well. Obviously, Robert Duvall's character is over the top. I doubt he would be in combat without a helmet, but there's a grain of truth there. It would have been more common for the commander to remain overhead to control the battle rather than land with the "Blues" (The infantry platoons of the Cav troops).

Rex Gooch, Ace: The Story of Lt. Col. Ace Cozzalio

Matthew Brennan, Headhunters