The Sawyer Mini plus a squeeze bag works well, but it's even better to have a system where you set up, go do other things, and come back in a few minutes to harvest clean water. Here's a mod to do that without having to develop some elaborate system. You can also see the system in a video.
The photo shows the bottom of a 32 ounce (950 ml) Sawyer squeeze bag, which I've adapted for gravity feed. Near each corner you'll notice an oval weld, in the plastic that isn't part of the actual water pouch (more like a flange). Rest the pouch on a piece of scrap lumber (not your hardwood cutting board) and, with the tip of a sharp knife, carefully pierce an X within each oval. Turn over the squeeze bag and repeat if necessary, until you have a small hole near each corner. Slip the hooks of a small, lightweight bungee cord through the holes to create a suspension handle. You're done!
To use the gravity feed system, find a branch where you can suspend the bag just above a wide-mouth water bottle, using paracord or other utility cord and a carabiner (you carry those things
anyway, right?). Fill the squeeze bag with water and screw on the Sawyer Mini. (You can also attach the provided straw to the clean end of the filter, to improve "aim.") Now suspend the bag and
filter upside-down, catching the dripping clean water with the water bottle. When you come back later, your clean water will be ready. Repeat as needed to replenish your water supply.
You can't use a hard-sided bottle as your source, because as the water drains a vacuum develops in the source bottle and defeats the force of gravity. The Sawyer pouch collapses as the water drains, so this problem doesn't occur.
In a kitchen experiment I inverted the full squeeze bag over a Nalgene bottle and timed the flow. I had my first cup (~ 240 ml) in 90 seconds. After 3 minutes and 45 seconds I had 2 cups of water, and after 6 minutes I had 3 cups of water. The gravity flow stopped after 9 minutes and 15 seconds, at which point I had 30 ounces (~ 890 ml) of water. (There was still a bit of water in the bag.) Clearly, the flow slowed as the amount of water in the bag decreased (that decreases the amount of gravity-induced water pressure on the filter). The experiment isn't an ideal one (sample size of 1, clean tap water in the "dirty" bag) but it indicates that the gravity-fed setup can produce a gallon (3.8 liters) of water an hour. And that's while you're mostly doing other things, such as eating lunch or setting up camp.