Here's one idea that seems to work (at home anyway). Should work in the field but one never knows. It only takes seconds to do it but it is a little tricky to explain with words and photos alone. Here goes:
(1) (assuming we start from the photo shown in the original post above) Remove stove and filter from bottle.
(2) Add a string to the bottle and add a bottle-end insert (cut from another bottle) to the group. See photo below:

(3) Stick your finger in the bottle's spout and scoop up some dirty water from stream or lake. See photo below:

(4) Shove the bottle-end insert into the bottle to hold water in the bottle. Invert the bottle and screw on filter. See photo below:

(5) Hang bottle. Leave bottle-end insert in bottle until you are ready to fill something with clean water. Gravity will do the rest. See photos below:


The rate of flow depends on amount of water in bottle. (i.e. more water = faster flow).