Last summer I had to find out myself how to get from AW to Tucson. There are three options:
1. The shuttles between US and Mexico.
2. A taxi to Lordsburg or Deming, from where public transport to Tucson or El Paso is available
3. Biking out.
There are several daily shuttles from AW to Phoenix or Tucson or Albuquerque. The disadvantage is that you have to wait in the middle of nowhere, usually in blasting heat, uncertain whether they have a free place. Find names of shuttle companies in
http://www.mataortizcalendar.com/traveling.html .
Sam Hughes from Hachita is willing to taxi riders from AW to Lordsburg or Deming. However, Sam is now 83 and had health problems recently. His telephone number is on the ACA map and his house is next to Hachita's Post Office.
Finally, the bike out to Tucson (3-4 days). I was told that the San Luis pass directly west of AW to Cloverdale is closed by the Border Police. So you have to return first on NM-81 to Hachita. Assuming that you don't want to bike the Interstate 10 to Tucson, because of the nauseous diesel fumes, the most direct route is NM-9 over Animas (good grocery ; Hachita's grocery is out of business) to Rodeo. Then to Portal, dirt road over Onion Saddle to Chiricahua National Park, Gleeson, Tombstone, Sonoita, Vail to Tucson. Or fully on paved roads: Rodeo - Douglas - Bisbee - Sierra Vista - Sonoita - Vail - Tucson. There is also an intriguing dirt road, rarely used by bikers, between Animas and Douglas called the Geronimo Trail. See
http://www.crazyguyonabike.com/doc/page/?o=1&page_id=310078&v=3D .