This is probably an easy question. I have some trouble finding the right notation/words for 2 vector fields.
Consider $T^*S^1$, the cotangent bundle of the circle. I know that this is a trivial vector bundle, so $T^*S^1 \simeq S^1 \times \mathbb{R}$. How do I formally introduce the vector fields on $T^* S^1$ that "goes around the loop" and the vector field that "goes along the cilinder"?
Any choice of coordinate chart $(U, \phi)$ on $\Bbb S^1$ (say, with coordinate $x$) determines a preferred coordinate chart $(\hat{U}, \hat{\phi})$ on $T^* \Bbb S^1$, where $$\hat{U} := \pi^{-1}(U) \quad \textrm{and} \quad \hat\phi(p, \alpha) := (\phi(p), q) \in \phi(U) \times \Bbb R ,$$ and where $\alpha = q \,dx$. (Here, $\pi$ is just the canonical projection $T^* \Bbb S^1 \to \Bbb S^1$ given by $(p, \alpha) \mapsto p$.) As indicated in the question, we can patch these charts together that give a canonical identification $T_1 \Bbb S^1 \stackrel{\approx}{\to} \Bbb S^1 \times \Bbb R$.
In the coordinates $(x, q)$, the vector field $\partial_q$ is vertical (i.e., tangent to the fibers, as $T_{(x, q)} \hat\pi \cdot \partial_q\vert_{(x, q)} = 0$ for all $(x, q) \in \hat U$) and the vector field $\partial_x$ is "orbital". If we choose $U$ to be an angular chart, then in the chart $(\hat U, \hat \phi)$, these agree with (under the appropriate identifications) the standard vector fields $\partial_q$ and $\partial_{\theta}$ on the factors $\Bbb R$ and $\Bbb S^1$ of the above factorization.