Very little becomes impassable in the winter unless it rains. Silt based mud is the enemy. While there are not a lot of direct off-road routes, there are a ton of trails in and around Tombstone and Bisbee. In the summer some of them can have some sketchy characters (camping; ie - Breaking Bad) out in the middle of no where. In the winter, not so much. Too cold.
Around Sierra Vista is Fort Huachuca. Busy winter Army base and some interesting topography around.
If heading west on I-10, Between Lordsburg and the AZ border (stop and get fuel), Hwy 80 parallels the NM/AZ border. I love this road. Hardly anyone around on the couple dozen times I have run it except BP trucks. It eventually crosses over from NM to AZ though there is no sign telling you. Cool abandoned mines and houses from the 1800's dot the area. Follow it all the way south and it takes you to the Douglas border crossing. Fuel, Walmart, food are there. About 3 miles from the border, Hwy 80 climbs west through the mountains and tailings from the copper mines that proliferated the area. Gorgeous scenery. Stop in Bisbee for a night. Locals can direct you to the trails N/NW of town. From there head out to Tombstone area for more trails. not a lot is available on maps, but they are out there. I've done them, but it was about 10 years ago.