New Mexico. The official deadline for Barbary sheep, bighorn sheep, deer, elk, ibex, javelina, pronghorn and oryx hunts in New Mexico, for the 2022-2023 hunting season, is March 19. However, New Mexico encourages you and other hunters to apply early. Those who apply before March 9 may have a chance to win a prize from sponsors including Kawasaki and Sportsman's Warehouse. As pronghorn antelope hunting season begins we find out from our hunters what hunting really means to them, and what they get out of it Our Antelope hunts are second to none and can be a.