Size is not an important factor in a reining horse as a rule as long as the horse has overall balance. The exception to this might be an extremely small horse not being suited to a large rider. Riders should wear whatever their tastes and budgets dictate. Think of it as a job interview. In other words your ability doesn't depend on what you wear, but it never hurts to make a good impression.
Straw hats are appropriate in h ot weather.
Felt hats are always in season. A nicely shaped hat gives the look of a serious competitor. Sometimes riders use bobby pins or apply hair spray or golf club grip cream to the front of the sweat band to make it tacky. Long sleeves are a must and weather would dictate wool vs. cotton. The shirt could be coordinated with other colors in the outfit, particularly the saddle blanket.