I have trouble visualizing what you mean.
Roughly speaking I'd see two ways to support the camera if the present bracket is insufficiently rigid:
The existing parts of your stand are in grey, the camera is the black shape suspended from it. Both red bits are the optional supports.
The vertical one would be a support that would make the setup very solid; I suspect this is what you're referring to. This might indeed involve some kind of threaded rod, although I doubt that's a very convenient option. A bar with a clamp of some sort seems more convenient.
The diagonal option offers an additional linkage towards the existing main column and should help avoid sag in the existing crossbar attachment. Again, some kind of clamping system seems the most appropriate.
Things will be easier if the camera is always at the same position e.g. for 'negative scanning' of the same format. If it needs to move, especially steplessly, it may be a bit more difficult to set this up, but it's still possible.
Implementation would also depend on whether you want to limit yourself to whatever you have in your parts bin vs. if you're willing to purchase or manufacture some additional hardware.