San Pedro Parks Wilderness is one of my favorites. Partly because it lacks the steep elevation gain of so many other NM hikes (sometimes a gal just gets tired of slogging UP) and it’s just gorgeous (there’s a reason there’s “Parks” in the name). Plenty of water — no need to carry more than a liter or so at a time. Lots of places to set up camp. Nice temps, wildflowers. Downsides are some marshy areas, even into July/Aug. It can rain — we’re not talking just monsoonal rains, but actual rain lasting hours (although, to be fair, I have a tendency to attract rain on backpacking trips, so that just might be me!). There are free range cattle who will give you the hairy eyeball (I’m pretty comfortable with livestock but bushwacked around large herds a couple of times). The cattle can make the marshy areas way worse by trampling & pooping. Last year I was there the weekend of July 18th & it was still marshy and churned up by the cattle. If you like to fish, bring your rod — there are at least a couple different kinds of trout and not many people fishing once you get a couple of miles from the trailhead. Good chances of seeing elk there (although my luck has been abysmal, limited to tracks & scat).
I have not backpacked in the Pecos Wilderness yet, but it’s great from what I hear. I’m headed there next month.
Also have not backpacked in Santa Fe NF, but have done some dayhiking there. It’s lovely, lots of shade, many of the trails have water access and you can link together some nice trails to alpine lakes. Trails are a bit more crowded than San Pedro Parks Wilderness but most are not bad, even on summer weekends.