Sub topics with Latent Dirichlet Allocation
I actually do not think your method is a good way to find subtopics.  Consider a document X with a distribution of topics z.  X is made up of a mixed model distribution of topic Z.  If you just give a document the most domiant distribution, and then run lda again, you might find subtopics but you'll also refind the topics that should should perhaps not be considered subtopics.

For example, let's consider a document that talks about food and exercising and why certain foods and cardo is good for heart health.  Let's assume that we find topics 1 and 2 such that the main topics are food, and exercise.  Suppose that the domiant distribution is topic 1.  Then running LDA on that particular document again, you're going to refine food, exercise and perhaps nutrition, viatims, etc.  However, there's no real reasons what should be considered the topic without inspecting each one and making an inference regarding it.  Also you'll have no real way to discern how deep you should go down this tree.

The original paper on hLDA can prove useful and exploits the chinese restraurant process and its relationship to dirichlet distribution to form the topic relationship and tree depth problem.  Also on David Blei github (bleilab) has an implentation in c++.  You can essentially setup a little shell script after processing the data in a langauge you may be more familar with and use his code.