Can I use an existing 6/3 cable plus an additional ground wire for a sub-panel?
No, when you retrofit a ground, you must source the retrofit ground from the same panel as the conductors come out of.
Further, such a groundless feed can be retrofitted for ground but it cannot be used in new work.
It sounds like the house was built with electric, and you went to a lot of trouble to convert to gas.  This is often done &quot;with a vengeance&quot; because traditional electric ranges are so awful compared to gas.  So your logic is &quot;Why NOT burn the bridge and make it impossible to go back, since any sane person will stay gas since electric is so awful&quot;. One technology change is inductive ranges, which by all reports are turning out to be quite good.
Another tech trend is toward electrification.  Given these sea changes, I would expect in 5-10 years a new homeowner may want to dispense with the propane altogether in favor of inductive range, extended range heat pump, etc.  This may be helped along by incentives either from gov't or power company.
Therefore, given that the 6/3 &quot;no ground&quot; feeder is utterly useless as a subpanel feeder, and still useful to the next homebuyer for fitting an induction range (and thus, adds value to the home), my advice would be to leave it alone.
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I note your your existing subpanel has a 2-2-2-4 feeder going to it. That feeder is good for 90A (if the existing subpanel is). So I would simply &quot;tee&quot; off that existing feeder with additional 2-2-2-4 to the new kitchen subpanel.  Now both old and new subpanel are sharing a 90A feeder.  That's more than plenty for a gas kitchen.