The Garden is one of the main trailheads used to access the High Peaks. I would suggest this as the best route and fastest.
Leaving the Garden trailhead, hike on relatively flat terrain through the valley to Johns Brooke Lodge (a headquarters in a way, where many many trails branch off from). From Johns Brooke Lodge (JBL), ~3.5 miles from where you parked your car, it is recommended that you find a campsite in there. There are many lean-to's in that area as well as designated camp spots. JBL is very used in the summer and fall so arrive early for a campspot, ESPECIALLY if your are trying to get a lean-to. I have found it to be nearly impossible to fine one and I have done about 30 of the High Peaks and am yet to get to a lean-to when someone wasn't occupying it.
From JBL, leave early in the morning and follow the Phelps Trail to Mt. Marcy. There are also signs for things along the way such as Bushnells Falls and Slant rock. Shortly after Slant rock, the trail branches left or right. Left goes to Haystack and right goes to Mt. Marcy. I should note that once you get to Slant rock, the rest of the way is very steep terrain, requiring the use of roots and trees to grab onto to pull urself up.
The route is a relatively standard hike, in my opinion, until you reach Slant Rock. After that it starts the climb. I would describe Haystack as one of the hardest mountains to climb in the ADKS. I would also say it has one of the best, if not THE best views in the ADKS. Another reason I favor it is that the summit rarely has many people on it. Everyone seems to do Marcy and on Marcy you can easily find 20+ people there on a nice day. I saw 3 people on Haystack when I did it on a very nice day.
After making the left to ascent Haystack, it is about .3 to .4 miles to tree line where you reach false summit #1. From there you can see Little Haystack and Big Haystack in all their glory. Continue on the trail to Little Haystack where the rocks will be spraypainted Yellow in parts to tell you where to walk. Follow the painted rocks down little haystack and up Haystack to the summit. From the first look out/ false summit, it takes about 30-40 min to summit Haystack. It looks really challenging and is but its TOTALLY worth it!!
Standard hiking gear..NO cotton!
Winter hikes, the ADK's require snow-shoes and winter approved clothing. that is it. (Although this hike would be extremely difficult in the winter since it is so exposed and the last .5 mile is scrambling on big boulders and rocks; it may be easy to lose the trail)
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