Use an existing 6/3 cable plus an additional ground wire for a sub-panel

Yes, you can. The systematic computer programmer in me even thinks it's a good idea, allowing a "star" topology for electrical distribution.

1. You will need 4 wires, as others have mentioned.

2. Keep the ground and neutral separate in the subpanels. The only place they should be bonded is in the main panel.

3. You will need ground rods at the new location. Yes, it's redundant, but it's still required.

4. The wire feeding the new subpanel must be protected from overcurrent by an appropriately-sized breaker. It can be at either end - in the old subpanel or in the new subpanel - or at both ends. It's up to you, but if only using one, I slightly prefer the upstream location, so that I can easily kill power to the feeder wire. If using both, 1 must be sized for the conductor, and the other can be that size or larger -- convenient if you buy a panel with a 100A breaker preinstalled.

5. While there are wires rated for direct burial, it's good that you're planning to use conduit. Oversize the conduit for ease of pulling wires, and minimize bends. While you have the trench open, run a second conduit for future expansion (ethernet? gas? macerator?). May as well run a water line, too, so you can have a sink.

6. Most electric kitchen stoves are installed on 240V/50A circuits. If you're installing something like that, then I suggest a 60A subpanel. That gives you plenty of headroom to install lighting, a small point-of-use electric hot water heater, and some convenience receptacles. Those will come in handy when doing maintenance in the area.

7. Aluminum wire is usually much cheaper than copper, but harder to work with. It's stiffer. You must use 1 size larger than with copper. The ends must be protected from corrosion with anitoxidant goop. The lugs must be torqued correctly. Some people think you should retorque the lugs on some schedule. Last time I bought wire, I wanted 8ga copper, but 2ga aluminum was 1/2 the price, so going with Al was worth it.

You are probably in violation of National Electric Code, article 250.32(B)(1) states that the grounded conductor(neutral), shall not be connected to the equipment grounding conductor(ground).

The general rule of thumb is the only point the ground and neutral get bonded together is at the first point of disconnect for any incoming system, be it utility, generator, or some other source. There are a few exceptions for specific situations but none of which I feel apply.

Easiest course of actions is to kill panel, un-bond bars, drive screw through one bar into chassis and make that ground bar, use the other isolated bar for neutral. With no other knowledge of your installation I would advise you be careful of the loose neutrals while making the swap, if any are somehow being used by circuits from the other panel(it happens) they could shock you.