Does Hamiltonian including spin involves tensor products?

I’m trying to learn introductory QM from a book and I’m confused about how spin is incorporated into the formalism.

From what I have gathered so far, the state of a particle can be fully described by two parts:

(1) the wave function of the particle as a solution to the time independent Schrodinger equation

(2) the spin of the particle

I keep reading that the "whole system" is described by the Hamiltonian, so I was wondering if in more advanced QM treatments, they consider the "whole system" to be a state in the tensor product spanned by states of the form $psi_n otimes | s, m_s rangle$ where $psi_n$ is an eigenstate of $H$, equipped with the Hamiltonian $H otimes I$ and time independent Schrodinger equation: $$(H otimes I) (psi otimes | sigma rangle) = Epsi otimes | sigma rangle $$