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If you're looking to beat the crowds, your best bet is to actually do backpacking in or around Zion. While Zion is very well known throughout the world, Kolob Canyon (which is a short distance NorthWest) from Zion, is a more hidden. The best part about Kolob Canyon is that the pass you buy there lasts for a week and it also includes entrance to Zion. So, if you get a ticket to Zion or Kolob, you have a ticket to both.
If you'd be willing to spend an entire week just dedicated to this region, you could take the trail from Lee Pass in Kolob all the way to Angel's Landing in Zions. The big issue is the amount of time and getting back to your vehicle after the hike.
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