A white linen pocket square is a great investment, because it can be worn with so many jackets, suits and of course, a morning coat.
Prince Charles in Brown Blue Patterned Tie with Stick Pin, Boutonniere & Silk Pocket Square
The fold is entirely up to you and can range from a plain TV fold to a flamboyant puff fold à la Oscar Wilde. Avoid silk for this purpose, as it cannot hold its shape as well and will be too shiny for the occasion.
Of course, you can also wear other pocket squares with the morning coat but when you wear morning dress for the first time, white linen will be a good start. In case you opt for something different, micro patterns in silk work well, especially if they harmonize with shirt, boutonniere or waistcoat. Due to the formal character silk or linen used to be the pocket square materials of choice. However, nowadays I can also see how silk blends, cashmere or wool may be a great option to create a more unique outfit. Of course, traditionalists should stick with either silk or linen.