This is another easy ramble, suitable for kids of all ages, of the kind so common in the Jemez Mountains. In the fall, if your timing is right, you'll get splashes of gold from the aspens. It's not as far into the mountains as Calaveras Canyon, so if you want to spend less time driving, you may prefer doing this one.
To get to the trailhead, head up NM 4 through Jemez Springs and turn left onto NM 126 at La Cueva. Follow that paved road until you're just past the turnoff to Fenton Lake State Park. Turn right onto FR 378 (at 35 deg. 53.497 min. N, 106 deg. 43.212 min. W) and drive 0.4 mile to cross a cattle guard in a north-south fence. Once past the cattle guard, pick a spot to park and start walking up canyon. FR 378 goes all the way up the canyon, ending at FR 144, but in places it's too rough for anything but a high-clearance 4WD vehicle. People in most vehicles should pull over and park no later than about 35 deg. 53.608 min. N, 106 deg. 42.291 min. W.
To do the hike, follow the forest road as far up canyon as you feel like, have lunch, then head back to the car. On the way up you'll pass a couple of side canyons you can explore if you wish. At 35 deg. 54.711 min. N, 106 deg. 39.908 min. W, the road you followed ends at FR 144. Continue another 90 meters east through the trees, and you'll be on the rim of San Antonio Canyon. The views are just glimpses through the trees, but it's a shady place to have lunch. Across the canyon you can see San Antonio Mountain (to the northeast) and Redondo Peak (to the southeast).
Depending on where you park after the cattle guard, this is a 6 to 7 mile (10 to 11 km) hike round trip. The cattle guard is at 7780 feet and the lunch spot next to San Antonio Canyon is at 8770 feet, so the elevation gain is about 1000 feet over 3 miles.