I would like to trivialize explicitely and in the simplest possible way the tangent bundle $T_pS^1$ finding a diffeomorphism $T_pS^1 \rightarrow S^1 \times \mathbb{R}$ whose restriction to each point $p$ of $S^1$ is a vector space isomorphism between the tangent space at $p$ and $\mathbb{R}$.
In order to do so I introduce an atlas on $S_1$ made of the two following maps: $$\phi_1(p) = \theta_1, 0<\theta_1<2\pi $$ where $\theta_1$ is the angle between the positive horizontal semiaxis and the radius at $p$ (i.e. the "phase" of $p$ if seen as a complex number), and $$\phi_2(p) = \theta_2, -\pi<\theta_2<\pi $$ with the smooth transition function $\theta_2 = \pi - \theta_1$. The map $\phi_1$ covers the open set $S_1 - (0,1)$ while $\phi_2$ covers the open set $S_1 - (-1,0)$. I understand we can give $T_pS^1$ the differential structure induced by the maps $(\phi_1(p),\phi_{1*}(v))$ and $(\phi_2(p),\phi_{2*}(v))$ where $v$ is a tangent vector at $p$ and $\phi_{1*}$ and $\phi_{2*}$ are the differentials of $\phi_{1}$ and $\phi_{2}$ respectively. This means that if I have a curve $\theta_1=k_1t+\theta_1(p)$ its velocity at $\theta_1(p)$ (a tangent vector at $p$) can be identified as $k_1$, hence either $k_1=\phi_{1*}(v)$ or $k_2=\phi_{2*}(v)$ be chosen as a coordinate of the single dimensional tangent space at $p$ where $\phi_{1}$ and $\phi_{2}$ are defined respectively.
For each $(p,v) \in TS^1$ I introduce the following function $D: TS^1 \rightarrow S^1 \times \mathbb{R}$
$$D(p,v)=(p,\phi_{1*}(v)), \text{ if } p \neq (0,1)$$ $$D(p,v)=(p,-\phi_{2*}(v)), \text{ if } p = (0,1)$$
I believe that $D$ is the diffeomorphism I am searching for, because it smooth at each $p$ and, in particular, also at $(0,1)$, because in an open set of $S^1$ around $(0,1)$ $D$ is identical to $(p,-\phi_{2*}(v))$.
I am saying this because given to $T_pS^1$ the differential structure induced by the maps $(\phi_1(p),\phi_{1*}(v))$ and $(\phi_2(p),\phi_{2*}(v))$, $D$ is by definition a diffeomorphism as it is continous and its pullbacks to the coordinate functions are the diffeomorphisms $(\theta_1,k_1) \rightarrow (\theta_1,k_1)$ and $(\theta_2,k_2) \rightarrow (\theta_2,-k_2)$.
Does my reasoning make sense?
Thanks in advance
Following your idea in the comments, we have that at each point $(x,y)\in S^1$ a basis vector for $T_{(x,y)}S^1$ is given by $-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}$ so we can take $\varphi: ((x,y),t)\mapsto ((x, y), t\left(-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}\right))$ which we may as well write $((x,y),t)\mapsto ((x, y), t(-y,x)).$ This is clearly a diffeomorphism that satisfies $\pi\circ \varphi=p$, where $\pi$ and $p$ are the projections from $TS^1$ and $S^1\times \mathbb R.$