I honestly don't know how you could shoot too far out at cottonwood canyon, the public access borders the river and you have to cross a lot of private land to stay on the path.IMHO the best shooting area I've been is North Wolf Creek out past Hillsboro. Proper shooting lanes. I recommend going early and going deep into the forest. The close in areas are nice but crowded. The place I discovered was probably a twenty minute drive into the forest and we set some targets out to 250 yards, set up clay pigeons on the walls and we had steel about 100 yards out.
On the eastside I'd recommend finding some BLM land (south of Corbett and the South Side of the Sandy River has a nice rope swing and swimming as a perk). Wildwood right outside of Sandy is nice.
Clackamas county has a bunch of shooting areas at gravel pits. The most famous and probably the closest(50 min) to Portland is Pit 36(the mile marker) right outside Estacada.
I haven't been yet but there's a spot outside St. Helens that looks interesting and then there's north of Carson, WA that I've also been told has some potential.
I'm going out to Cottonwood Canyon (2 hours from Portland) looking to do some distance shooting in the near future, I'll probably do a write up on it when I get back.
Good luck, be safe and don't use up all of your ammo.
Let me know how it works out for ya, but I'd pack a pistol just in case...
Edit; Looked at a map, seems the land acquisitions went through and you might be able to get a spot near the big eddy. Good luck!
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