For finetuning on additional data, you would have to lower the learning rate because you're only adding a few datapoints and it's almost entirely converged as it is. If your loss is increasing, that suggests overfitting to me via a too-high learning rate
Now, if you're changing the architecture (such as by adding additional categories of pieces), as I said, that's more tricky - what people usually do there is something like lop off the top layers and retrain them from scratch, possibly while freezing the rest of the NN (the assumption there being that the learned filters and lower layers ought to already be sufficient to classify a new category, which is reasonable since the lower layers tend to be learning things like lines and corners, all primitives which should be able to classify yet another square or rectangle etc).
Since this is the obvious response any reader familiar with deep learning would have while reading complaints about how slow your CNN is to train from scratch, it'd be good to discuss it in some detail what sort of finetuning you've tried and how it failed.