How Do I Wire My Nest Hello Doorbell With A Plug In Transformer?
Make sure your new transformer has enough power for both your chime and the hello. I would not continue to use the old transformer. I would disconnect the power to the old transformer and rely on the power of the new transformer to operate both the old chime and the hello.

It looks like your old transformer is connected at the "0" and "3" terminals. Remove those and cap them off. Connect your new plug in transformer there.