So I will call the Poincaré bundle $(FM,\pi, M)$ the principal fiber bundle that has the Poincaré group as a structure group, the space of linear frames as total space and M as the Riemann-Cartan space.
I didn't find a book or a paper that treats this kind of principal bundle (I would be happy if you share resources that you found) so I did my own point of view on this typical example and I want you to correct me if I dismiss some things in my line of reasoning.
So the idea is that we have a Poincaré group (the restricted one) $\mathcal{P}^{\uparrow}_{+}$ and we find that this group is too big because the goal (for me) is the representation of the restricted Lorentz group (RLG) in a spinor space so we restrict the $\mathcal{P}^{\uparrow}_{+}$ to the RLG, so to do that we know that the RLG is a Lie subgroup of $\mathcal{P}^{\uparrow}_{+}$ (which is also a Lie group) and with the Cartan theorem on closed subgroups, we have as a result: the RLG is a closed subgroup. Nevertheless, we still need to search for a global section for the $SO(1,3)^{\uparrow}_{+}$-principal fiber bundle, the obvious one is the section that associates $\forall x \in M $ a section $ s: M\longrightarrow OFM$, $OFM$ being the Orthogonal frame space. Consequently, we restricted the Poincaré bundle to the $SO(1,3)^{\uparrow}_{+}$-principal fiber bundle.
Searching for a representation to this group, we can use the principal bundle morphism that is $\rho$-equivariant with $\rho: SO(1,3)^{\uparrow}_{+} \longrightarrow Spin(1,3)$ and with the representation space being the spin frame bundle, after that, we associate naturally a spin bundle with typical fiber $F \cong \mathbb{R^4}$ by virtue of a linear left action of the Spin group that we can construct from the Clifford algebra or just simply by a linear representation of the $SO(1,3)^{\uparrow}_{+}$ into the space of 4-by-4 matrices $S(\Lambda)$ that acts on spinor fields.
Then we can define the one-form spin connection, soldering form, torsion, curvature and the covariant derivative of a spinor field (section of the spinor bundle).
Thank you for your attention and just for you to know that am not trying to be extremely focused on rigorous step-by-step reasoning, but you are free to give me critics.