Alamo Heights is very nice, and the school district can't be beat if you have kids (although both the NEISD and NISD are very good as well). It's relatively expensive, so you'll blow through your BAH pretty quickly. I would consider Alamo Heights only if you either 1) have a second income or 2) don't have kids and don't plan on having them while in SA. The only downside of the commute is that most of the quickest routes to SAMMC from Alamo Heights involve cutting across FSH, meaning you'll have to deal with gate traffic four times for each daily commute.
To clarify, Alamo Heights is both a town and a school district, the latter of which comprises the town of Alamo Heights as well as part of Terrell Hills, Olmos Park (the tri-cities), and San Antonio. Most people mean the school district, because there isn't an appreciable difference between Terrell Hills in the AHISD and the town of Alamo Heights, and because cost tracks most closely with the school district.
The Army people I know mostly live across the north half of the city, mostly north central (Stone Oak) out 281 with a smattering in the northwest (out I-10) or northeast (out I-35) areas. Under no circumstances would I consider living out I-35, as that highway is a parking lot for huge stretches of the day even without accidents, of which there are many. The medical center area is mostly apartment living; plus, it's super shady around there. Do not want.
AF people are more likely to live on the west side (out Hwy 90) closer to Lackland, which makes some sense. But if you're Army and/or you'll be spending most of your time at SAMMC, then I'm not sure I see the appeal of living that far away.