How to use neural network's hidden layer output for feature engineering?
TL;DR:
Yes. You can (iiuc)

Longer Version:
In fact, this is what many popular algorithms like Word2Vec and AutoEncoders do. (With respect to hidden layer outputs)

Word2Vec:
Given an input word ('chicken'), the model tries to predict the neighbouring word ('wings')
In the process of trying to predict the correct neighbour, the model learns a hidden layer representation of the word which helps it achieve its task.

Finally, we just remove the last layer and use the hidden layer representation of the word as its $N$ dimensional vector.



So basically, we feature engineered the word vectors.

For autoencoders,
It takes in $X$ as the input and tries to predict $X$ again, in the process learning a latent representation of the input signal X. The input hidden representation in Layer $L2$ can be used in other tasks. (Note: here X and X hat are the same)


(In fact, you can use features learned by a CNN and feed them into an SVM and get good results)

Relevant to you:
You can train your model on the given data and finally chop off the last prediction layer and use the output of the intermediate layers as features. I believe this should work because it works for so many tasks which I explained above. 

Source:


Learn Word2Vec by implementing it in Word2Vec : an article by me explaining word2vec. (Shameless self-advertising here but I feel the article is good and relevant)
Andrew Ng's unsupervised feature learning website