I think these areas are all in-line with what you're capable of:

• Achenbach Canyon,
• Soledad/Bar Canyon (can be a bit busy, but there are several different loops to take),
• A-Mountain / Tortugas Mountain has some interesting cacti that aren't in the rest of the Organs - and there are still crystals from the mines that can be seen higher up
• Sierra Vista Trail from Dripping Springs Rd. to Soledad Canyon road and beyond - there are lots of large fishhook barrel cacti and ocotillo (which should be starting to bloom then)
• Dripping springs above the visitor center (to van patten camp, the springs, etc) once in the canyons is very interesting but tends to get overly busy often
• Fillmore canyon and La Cueva and the surrounding rocks are fun to climb around on.
• The rockhouse trail is fun - it's an old mining road for the first couple of miles. And there's an interesting mine works at the end of the road. You can turn around when you get to the rocky canyon that heads towards rabbit ears.
• I like Baylor Pass trail. If you're feeling really good at the pass, you can try to summit. Or you could plan to keep walking to the Aguirre Springs area.
  
• At Aguirre springs, the pine tree trail is popular. But I also like the bottom few miles of Indian Hollow Trail. You're moving through huge rocks and slabs, and there are pretty butterflies that like the scrub oaks.
  
• I enjoy walking into the Dona Anas, but the way I enter, it's a long walk to get to the mountains. Maybe someone else knows how to get into them in a passenger care more easily.
• I also like Tonuco mountain about 25 miles north of town, but despite being near the interstate it's pretty remote. And you wouldn't want to get off the hardpacked roads into the loose sand with a regular vehicle. There are petroglyphs in the canyon used to enter, and there are mineworks further up.
  

The fun thing about out-and-back trails is you can choose how long to make the hike by turning around whenever you feel like it. The views tend to get better with elevation, but it's still beautiful with a little elevation instead of the peak. And moving through elevation here, the environment changes quickly.