I've used Transtint dyes to color clear oil finishes (tung oil, boiled linseed oil). The dye is supposedly water-based but it's got some other stuff in it to make it more or less soluble in oil. A few drops go a long way, but I suppose if you try to use too much dye the oil won't take any more. You can affect the top color by using your dyed oil on top of existing finish. Prep the existing finish as you would for any top coat. Prep is important or the oil won't dry and cure properly. You might need to do multiple coats. It's likely to be a lengthy process. As mentioned above, experiment first, if you can, on a spot that can't be seen.
I'm pretty sure both stain (penetrating or gel) and also aquafortis won't work unless you first completely strip the existing finish. The varnish stain mentioned might be a less time and work option. I don't think it gives as nice a finish, as oil, though. But do what works best for you.