You're trying to add another dimension of label, not a feature. You need to edit each of your input files to contain these separate classes (0-L through 9-L, 0-R through 9-R), and then scale up the topology to handle the added discrimination.
Scaling the model will take a little experimentation. I suggest that you begin with 20% to 100% more nodes in each hidden layer; run a series of experiments to find your optimum topology.
UPDATE after OP comment
Zeroes also lean left and right, but it's often more subtle. However. the main reason is that you're making a more detailed set of decisions, so you're likely to need more intermediate factors. These are the hidden nodes.
Try the original topology on 20 classes; see how well it works, and start from there. I think you'll need more nodes, but the ultimate authority is your trained model.
Building models is still something of an art, although we're learning more every year. Where the problem is very similar to one with a known solution (i.e. yours), we can often get a good solution by altering the "width" of the current model to compensate for the difference in problems.
In your case, twice as many classes to discriminate suggests a simple increase in the quantity of hidden nodes. However, whether that increase is to double the nodes, increase by a smaller amount, or merely alter the final layer from 10 to 20 classes, is an exercise for the deep learning researcher -- that's you. :-)
Another UPDATE
I see -- it is input, not a classification. My error, not yours.
Yes, you can do that ... but you'd need to change your input format to accommodate the added information. This means adding a bit to each input file to signify the hand, and reflecting that extra bit in your ingestion method. I expect that the model will adapt with the current topology.