As the highest point in the frontal range along trail 119, Hershberger Peak qualifies as a destination hike, but other than taking you to that obvious aspect, once there you'll recognize it as a fairly overgrown and undistinguished point. It's shorter cousin to the north, Ortega Peak, has a much nicer 360 degree vantage, and a more pleasant vista. Still, it is the highest and a peak to boot, so if you must... go for it.
The access is simple for the first miles... follow trail 119 ("A" Mountain Trail) from it's lower end down in Alamogordo, or from it's upper end on West Side Road (FR 90) up to the point north and east of the mountain. If you are hiking from the lower end the tendency will be to want to exit the trail too soon. Don't. The watershed to the north of the mountain (the drainage that will be on your right as you ascend) cuts completely up past the mountain. You need to keep going until you are well past the mountain and crossing over the ridge to the drainage to the south of the mountain (Cabellero Canyon). When you have reached this point you exit towards the mountain. There is an obvious ridge and small hill. Past that is an obvious false peak. It might seem plausible to skirt that peak around to the right and not waste the elevation gain/loss, but really it is easier to simply follow the game trails right up and over the interim peak. The best access lies right on the ridge point on the south of the false peak, and then staying with that ridge line, dropping down into the saddle below and connecting to Hershberger. The best ascent up Hershberger is also to stay as far to the southern edge of that ridge as possible. Some of the game trails are very clear, especially when the going gets extremely steep. If in doubt, move further left and continue to climb. There is no trail, but elk and deer have worn down paths nicely enough to provide fairly clear access, especially if you are no taller than 4' 6" (it would be nice if taller game would begin to cut and clear taller trails... ). Some of the paths may require a bit of scrambling up and over rock ledges, very sharp, volcanic rock. Lots of Spanish dagger, cacti, and shin daggers for potential minor injury. There are very loose sections of scree, nice for a bit of downhill sliding, ideally on the way back down.
The peak is overgrown, but has a geological survey marker (Alamo Peak) and a memorial dedication.
Again... it's a peak to bag, but for more fun, hit Ortega and spend time there. Look for the old mine. Enjoy the views.
To hike In Alamogordo, from the stop light at the intersection of 10th and White Sands Blvd (the main drag thru town) turn east towards the mountain and follow 10th up until it ends, curving to the left at the fire station (Pauite rd). Drive a tenth of a mile and park next to the curb just beyond the Sunrise Ave intersection. (I pull up on the vacant lot on the right just before the culvert wash). Walk east towards the mountain, and find the marked trail heading up the hill. Continue up, ignoring the many atv cuts in the lower area. It is an old jeep trail and is fairly easy to follow.
WARNING! Hiking and outdoor related sports can be dangerous. Be responsible and prepare for the trip. Study the area you are entering and plan accordingly. Dress for the current and unexpected weather changes. Take plenty of water. Never go alone. Make an itinerary with your plan(s), route(s), destination(s) and expected return time. Give your itinerary to trusted family and/or friends.