From what you wrote, no, you cannot do that so easily.
Your transfer switch is a 10 circuit. Let's assume that 6 of those circuits feed loads in the "new" section of your house, and it would be convenient for you to move the transfer switch to the garage to simplify the wiring of those 6 circuits. I guess that's the impetus for your question?
So far so good. You can install a transfer switch in the garage, connect your generator to it, and power those 6 circuits either from the generator of from the garage sub-panel. Minimal wiring, as the generator is next to the garage.
But what about the remaining 4 circuits from the current transfer switch near the main panel? You cannot power them through the existing 6-3 wire (back-feed).
You could wire a second wire between the main panel and the garage sub panel, to carry the generator power to the existing transfer switch. You would have 2 transfer switches, in parallel.
Another option would be to move the transfer switch to your garage and install extra wires to carry those circuits (4 in my example) back to the main panel. If your transfer switch only switches the hots and not the neutrals, then you need to wire only the hots, for example as individual wires in an EMT conduit.