To succinctly answer your question...
Wimberly makes a product line called the "Plamp", which is primarily designed to hold more delicate things than jewelry (flowers that one would not like to damage, etc), but would also work well for jewelry, or any small objects.
However, there are some other options you may want to look at:
A 360 Product Turntable
...with any kind of small alligator/macro clip should work nicely. You could also, as some suggested, hot glue the item to a small card, a tiny foamcore square, or anything like that.
If that sounds like your solution, check out Orangemonkie's Foldio product line
It is designed for 360 product shots, is motorized, has its own lighting, and is controllable with a smartphone. Specifically, take a look at the Foldio 360 turntable and the Foldio Studio (a diffusion "tent" used to even out your lighting).
Regardless of your solution, I would honestly advise you to get some sort of surround diffusion for your lighting, since evening out the light will eliminate a lot of the hurdles you are currently jumping to get your models.
There are dedicated turntables for 3D, but they can be quite expensive ($2000-3000+).
Another option, using a different strategy...
If you're not completely attached to your current workflow for some reason, another option is the Intel® RealSense™ Depth Camera D415, which can scan small objects in just a few passes (using infrared grid projections, stereo cameras, and other fancy tricks). It could certainly accomplish your task in much less time.
Disclaimer: I haven't used one of these.
One other gadget that may help you,
should you decide the DSLR is the way you want to go is a focusing rail for your camera support (it's easier to move the camera sometimes). I have used the Oben MFR4-5 Macro Focusing Rail to take some excellent flower shots, and it's one of the few dual axis rails you can get for less than $100.