I'm quite new to hammocking after decades of being a basha "ground sleeper" and I only carry climbing carabiners when climbing - which I rarely get to do these days. I generally carry (EDC) one or two 5mm steel snaplinks or 6mm (chain) steel quicklinks. Quicklinks are loadbearing, used in freefall parachutes between the riser and rigging lines (even on military HALO/HAHO rigs).
I've devised an EDC pouch which contains my hammock and a poncho. Devising, testing and reworking it has been a real joy: I think I have it as close to perfection as it can get. I made the hammock from ripstop parachute fabric, the poncho is an unbranded lightweight imitation of the old '58 pattern poncho (charity shop find). My hammock fits into a '44 patt mug, the poncho a tad bigger.
Aiming to be uber lightweight and compact, I dispensed with metal links to attach the hammock to, instead looping each whoopie sling around a marlin spike hitch on the tree strap. The ONLY drawback so far is that, after loading, the sling can bite into the knot (not in a damaging way) and cease to slide easily behind the knot without first being loosened.
To get around this, I looked up the current gadgets on the market and found "Dutch 'Biners" to be popular. But since these look largely like open 6mm quicklinks, my next test will be using 6mm quicklinks. I don't think I'll even need to close the gate. I like the idea of every item of kit that I carry serves multiple purposes - which means that I'd NEVER buy/carry kit like Dutch 'biners, not that I'm knocking them.
I'd be very interested in the opinion of more seasoned hammockers.