These are some of the routes I've been considering:
SA to Flagstaff
To OEW 1 (Highlights: Roswell, NM; Valley of Fires Rec Area; Very Large Array Observatory)
Hwy 285, Carlsbad, NM to Roswell
Hwy 380 Roswell, through Carrizozo and the Valley of Fires, to I-25 up to Socorro
Hwy 60 W from Socorro, past the Very Large Array, to Springerville, AZ
Hwy 191 to Hwy 180 (St. John's) to I-40 (Holbrook)
I-40 to Flagstaff
To OEW 2 (Highlights: Coronado Trail - considered a "classic" scenic byway)
I-10 to Lordsburg, NM
Hwy 70 to Duncan, AZ
Hwy 75 to Three Way
Hwy 191 (Coronado Trail) to Springerville, then to Flagstaff via I-140 as before
Flagstaff to SA
From OEW 1 (Highlights: Gila NF and Gila Cliff Dwellings)
I-40 to Holbrook, AZ
Hwy 180 to Hwy 191 to Springerville
Enter Gila NF via Hwy 60 and Apache Creek or continue south to Alpine and take Hwy 180 to Luna
Once GNF there are several gravel and dirt roads accessible from Hwy 180 and Hwy 12 that cross the Tularosa Mts and eventually connect to NM 59, which also intersects a Forest Road that leads to NM 35. From there you can turn back north to see cliff dwellings or continue south and out of the park
A map of Gila NF is available
here. Lists a lot of roads that you don't see in Google Maps. There's a lot of options for exploring, beyond what I mentioned.
Time will be a factor for me. San Antonio to Flagstaff is about 15 hours so while it's doable in a day, spreading it across 2 would be preferred. Traveling through Gila NF is possibly the biggest unknown. Not sure how long it takes to cross and you can always make the time spent there longer with the more exploring you do.
Ideally, I think I'd like to leave early Wednesday, from SA, and arrive in Flagstaff Thursday afternoon. 3 days of expo and hit the road late Sunday. Be back at work again Wednesday or Thursday. I've even considered not staying for the entire thing but it seems like you lose access to a lot of good classes if you don't register for the full expo. Balancing family, work and time away will be key for me and play a large role in how much time I spend on this trip.