I think the knot you're using is appropriate if the idea is not to lose your kit, another knot may run you the risk of losing something because it came untied without your blessing.
If you're using cheap cord, then don't bother with the knot if it's giving you grief, just cut it at the knot, your cord will get shorter each time, but it still usable for a couple cuts.
If you want to be able to attach something to your bag or belt quickly without fiddling with your knots, then I recommend simply girth hitching it:
Keep your loop tied to whatever with a double fishermans, then you can girth hitch it to practically anything as long as you can fit the loop around. In the image above I tied the cord around my strap, then passed a loop through the basket and folded it back around; only takes a second. You could even girth hitch both ends, then you don't have to tie cord around everything, just keep a bunch of tied loops, this makes it easy to use with a small carabiner too.
This answer assumes that you're trying to attach something to a loop (the black and green band in your photo) that you can't open up. If you can open up the loop, then see ShemSeger's answer.
It depends on how quickly you need it to release, and how important it is to you for it to stay secure.
A secure knot that is pretty easy to untie, even after it has been loaded, is the offset overhand knot. I have had my keys in a loop of cord tied with an offset overhand, and after a year or two, the knot is still there. For good security, make sure to leave long tails, and dress and tighten the knot carefully.
If it's important to you to be able to release the knot very quickly, you could use a slipped square knot, i.e., the one you use to tie your shoes.