Two of my friends and I went to Guadalupe Mountains National Park last winter to accomplish something we hadn't been able to do two years ago - climb Shumard Peak. Last time, we had tried via Bartlett Peak from Bush Mountain, but time ran short and my friend fell into an agave plant. This time, after studying the map of the park, we saw a pretty viable route that left Pine Springs Canyon not too long after Devil's Hall and went up one of the many ridges on Shumard's eastern flank. It worked like a charm. After leaving the rocky canyon bottom a little too early and having to traverse into the next gully over, we were soon on the right path. There was some bushwhacking, of course, but it was not terribly difficult.
Looking from Guadalupe Peak, 3 ridges seem to emanate from Shumard's summit - this is the middle of the three that we ascended. After climbing out of the drainage that runs between Shumard and Guadalupe Peaks, we gained the ridge and walked up to Shumard's false summit, and then finally over to the true summit. The route-finding is not very advanced and the biggest difficulty is in avoiding the sharp desert plants! It was great - there was virtually no sign of people being up there. Time from Pine Springs to Shumard's summit: 5 hours.
It was loose in places, but mainly leaving Devils Hall. After that it wasn't bad. We actually went over to Guadalupe Peak after that, so I'm not sure about the route to Bartlett...didn't look too bad that way. Good luck! Guadalupes are great