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Consider a sequence of tokens. One could fit a Markov model to this sequence by computing the empirical stochastic transition matrix.

A refinement to this model would be to build a HMM on top of it, with a small tweak: we let the emission probability of a token depend both on the hidden state and on the previous token. This is still a hmm but with a structure constraint.

We then take the Viterbi path of the hidden states, and fit an HMM on top of it.

And so on, with many HMM on top of HMMs.

Of course, the end result is still a HMM, but hopefully one with a lot of structure and easy to fit.

One example would be a natural language text. The first layers of HMM would describe common phonemes, higher levels common suffixes and prefixes, then words, parts of speech and sentence structure.

This is not a hierarchical HMM. Does this have a name?

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The model in which the observation at time $t$ is also dependent on observation at time $t-1$ is used in Named Entity Recognition with Character-Level Models (Klein et. al). The second model you describe is similar to Layered HMM.

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