How long do you need the smoke for and do you need it to "hug the ground"? If not some of those super large military smoke sticks could work wonders. I have six of 'em sitting down stairs waiting to be used on a short I'm shooting next week.
The fog effect i'm looking for is sort of like the opening scene of E.T. That kind of atmosphere is amazing. The longest single shot will be about 30-35 seconds. I don't want the fog to hug the ground.
Is it a pine forest? Are you allowed to have fires?
We used to start a small fire, then pile on a bunch of green pine boughs. They would smoke (with no flame) for a long time, and fill the forest with great "X-Files" smoke/fog.
Cheap, but you (and your equipment) will stink for a few years afterwards.
Also, this was in Canada, either on private property (where we had permission) or Crown land (where you don't need permission), so YMMV. Keep a big container of water or a fire extinguisher nearby in case it goes out of control (never happened to us.)
Now there's a suggestion for Kirador.
Come to Canada and start a forest fire.
Smoke machine,---who needs a smoke machine when you can start a forest fire.
But make sure it's on Crown Land!!!
I've used (rented from an effects tech in Baltimore) who used an IGEA (?) fogger for outdoor scenes requiring fog/smoke. The fogger is basically a converted pesticide sprayer converted to burn mineral oil. It was awesome !! I've used a few types of smoke machines (for indoors) and cookies and knew that neither of those would cut it for the outdoor/nighttime scene I was taping at the time.
It worked great and only needed to sprayed for only a few seconds to cover a huge forest. What's amazing is how well it worked during the day too (even though I didn't tape any daytime scenes with it). Any old fogger will look decent backlight at night but to get a nice fog effect during the day is a harder feat. It also runs on a 2-stroke engine so you don't need electricity (but it's noisy as hell, so fog before rolling).
You should check to see if a local fx person uses that type of fogger. I was in PA and had to drive to B-more to rent ours.
You can check out footage of the scene I shot for a historical doc. I was producing for PBS on my reel at: