It seems that most Bayesian inference focuses on inferring the posterior. Is it possible to infer both the prior and the posterior.
Your question is ill-posed, it doesn't make sense to "infer" a prior.
Let's say you have a likelihood $p(x|\theta)$, where $x$ is the data and $\theta$ are some parameters. In bayesian inference, the objective is to find the distribution of the parameters given the data, $p(\theta|x)$, which is the posterior. In order to do this, you first posit a prior distribution over the parameters $p(\theta)$. We then have from bayes rule that $p(\theta|x)\propto p(x|\theta)p(\theta)$.
"Inference" is usually reserved for the process of using data to learn something about the parameters of a model. Note that the prior doesn't involve the data at all, so it doesn't make sense to talk about inferring it.
Also note that empirical bayes isn't really related to reference/Jeffreys' priors (not Jerry's priors). In an empirical bayes setup you already posit a prior distribution, and use the data to set the hyperparameters of this prior. The point of reference/Jeffrey's priors (and objective bayesian inference) is to construct a prior using just the assumed likelihood.
What you're describing falls under the general rubric of "empirical Bayes" methods. These can be used when you have a large set of parameters that you have reason to think might be drawn from a single prior distribution.
An example from biostatistics: Suppose you have identified several thousand SNPs (single nucleotide polymorphisms, a type of genetic variant) in the human genome, and you want to makes inferences about their frequencies in a particular population. You can model these frequencies as being drawn from, say, a prior with a beta distribution, and you can use the observed frequencies to estimate the parameters of this prior. You can then use the observed data for each individual SNP to update this prior.