Let $E$ be a smooth vector bundle over a manifold $M$, equipped with a connection $\nabla$. Let $\sigma \in \Gamma(E)$. Consider $\sigma$ as a smooth map $M \to E$, and suppose that $(d\sigma)_p=0$, where $d\sigma_p:T_pM \to T_{\sigma(p)}E$ is the standard differential of $\sigma$ at the point $p$.
Is it true that $(\nabla \sigma)_p=0$?
Note that I am not assuming that $\sigma(p)=0$. If we do assume this, I think the answer is positive as can be seen from the local form of the connetion:
$$\nabla \sigma =( d\sigma ^\alpha+\omega^{\alpha}_{\beta }\sigma^{\beta })e_{\alpha}.$$
(This easily implies that for two sections $\sigma_1,\sigma_2$, if their first jets are equal at $p$, so are their covariant derivatives at $p$, right?)
It's not possible for a section to satisfy $d\sigma_p=0$. That's because the projection $\pi\colon E\to M$ is a left inverse for it: $\pi\circ\sigma = \operatorname{Id}_M$. This implies $d\pi_{\sigma(p)}\circ d\sigma_p = \operatorname{Id}_{T_pM}$, which implies $d\sigma_p$ is injective for every $p$.