You have a few options here. Of these, I think 1 will be the easiest to implement, as it's a standard language model with an alignment term added to the loss. I'd recommend 2a if you think you have the time, as I imagine its performance might be much better.
1 Use existing corpus to learn embeddings for new words
You can do this as you describe, though I would initialize the unknown embeddings with random noise instead of zeros. This will probably do fine, but it suffers from the problem that the model has no way of knowing that the SENNA embeddings are to be trusted more than the randomly initialized ones. So, it will generally take more (time, examples, etc.) to train this one well.
Another option is to try to capture the difference between known embeddings and randomly initialized ones. My suggestion would be to create your own embeddings [20k x emb_dim] and initialize it however you want. Then add a penalty to your model's loss for deviation from the SENNA embeddings. This is what's done in bilingual embedding models like this one. This will push the known embeddings close to SENNA and allow the unknown ones to freely vary. You could also reduce the coefficient for this alignment penalty as training progresses.
You could either learn these embeddings on a dedicated embedding task (like CBOW) or as part of your primary task.
2 Hybrid model for unknown embeddings
A second approach is to forgo attempting to learn embeddings for your unknown words at all. Typically, mosty word RNN models do this for low-frequency words through something like the <unk> tag. For you, that would be half your vocabulary. Fortunately, there are a lot of tools for filling in these out-of-vocabulary (OOV) words.
2a Subword embeddings
As @Emre pointed out, subword embeddings have been used to great effect to solve out-of-vocabulary problems. Here's an example of a model that uses subword embeddings to achieve an "open vocabulary" model. This means you would combine your pre-trained embeddings (which you would not need to make learnable) with a character-level model to learn when and how to produce OOV words.
2b <unk> replacement
Another approach is to, as a post-processing step, replace the <unk> symbols with words from your corpus. You can build your own language model (even a simple markov chain would be an improvement). Also, some modern models have more sophisticated methods for replacement that work quite well.
