here is an example on how to generate synthetic data that you can adapt for your case https://dzlab.github.io/2023/09/22/palm-synthetic-data/
Without seeing some kind of demonstration otherwise, my feeling is that it would be like regressing stock price on inflation, then trying to generate more data using the regression model and random inflation numbers. All you'd learn is the model that you put in to generate the data.
Even in the main article here, the model did better with fewer fine tuned examples. To us, the auto-generated examples might look different enough and might look good enough, but they were all generated algorithmically. Feeding more examples in might easily be leading it to focus on some artifact of the embeddings or generating model that we just don't perceive.
If you have a small student model and a large teacher it makes sense, the student is better off after this distillation.
If you have a way to filter out low quality synthetic examples then it would be useful to generate a bunch more and take the best.
If your LLM is an agent, then it can generate feedback signals from the environment. Even a human-AI chat is a form of environment for the model. Every human response can be evaluated as positive or negative reward.
More fundamentally, organic datasets are very unbalanced, LLMs need more complex reasoning chains than what is usually available. There are some exceptions - in scientific papers, manuals and code you get very complex reasoning chains. But not in general. This issue can be fixed with synthetic data.
And even in principle, if you have a model at level N and want to make a dataset at level N+1, then you need to boost your model. You can give it more tokens, more attempts or more tools.