I am not sure if this will help, as I don't use a JB, but I do use several different stoves and a key factor for me is that the water is no longer boiling, but has cooled off for a minute or two. Do you have another vessel that you could use, or do you drink straight out of the JB pot?
I use several Coffee items that I find I won't go into the BC without.
1.
Coughlins Filters - Ultralight weight and good for a 20oz mug of strong coffee. I don't think that in my coffee drinking backpacking life, I have ever taken a trip without having these little buddies in my pack.
2.
MSR MugMate - Good for a very large mug of coffee. More Fragile and more expensive, but allows some of the finer silt to come through, giving you that french press taste. I find a smaller (2 oz?) Nalgene bottle fits inside the screen so it doesn't get crushed in the pack.
3. Note: This probably won't work with your JB, but for any other reason......
French Press
All joking aside, I received this as a gift a few years ago from a (obviously) well meaning non-backpacking friend. I tried it out on the kitchen stove several mornings and then threw it in my pack
instead of a pot for some single overnighters and It was really awesome. I was absolutely impressed with it - It functions as a stainless steel pot, not much heavier than my old MSR cascade cook-set, that I can cook in, if I need to (Carry the larger lexan spoon) but for most of my needs, it is boiling water to rehydrate foods.