B&H still has heat sealed cardboard mounts brand new, and they are the cheapest option at 1000 for $40 but they're slow to mount. The only other new option I'm aware of are the AP Pro CAM/CS II style mounts. AP Pro, Agfa, Kaiser, and Kindermann all made compatible mounts, and they're pretty fast to mount with the dedicated slide mounter. The difference between the CS and CS II is just which direction the mounts open (long vs short side), but as a result, you'll need a different mounter for each type. Easiest just to go with the CS II style.
The Hama DSR is a pretty nice system, but as you've noted, you're stuck with finding whatever mounts happen to show up used. Both the DSR and the CS II are under 1.8mm thick, as AZD noted, so they'll fit in the thinner trays.
Both Gepe and Pakor made two styles of glassless plastic mounts, ones that simply press together or pry apart to mount, and ones that require a machine. The mounts designed for machines cannot be used by hand, as the machine either melts or deforms rivets to lock the halves in place. At this point, given that the mounts are no longer available new, it's not worth tracking down a machine to use the mounts unless you come across a case of a few thousand mounts.
I've been told by a few different labs that plastic mounts (besides the CS II style) are not being produced any more, by Pakor, Gepe, or anyone else. Cardboard mounts I expect to be available for the foreseeable future, as they're much easier to manufacture than plastic ones.