Let $M$ be a smooth manifold, $E \subset TM$ a smooth distribution of codimension $k$, and $\gamma : [0,1] \to M$ a smooth curve whose tangent vector is in $\gamma^\star E$, and whose individual points are contained in integral manifolds of $E$. Can we deduce that $\gamma$ is entirely contained in a single connected integral manifold of $E$?
If $E$ is integrable, then the answer is “yes”, and I can prove it as follows. For each $t \in [0,1]$, take a distinguished chart in which $E$ is defined by $dx^1 = \dots = dx^k = 0$. Then $\gamma$ is locally contained in a single plaque of this chart. Then we have an open cover of $\gamma$ by plaques. Take a finite subcover, and their union is a connected integral manifold of $E$ containing $\gamma$.
But what if $E$ is not integrable?
EDIT: Rewrote whole question.
Never mind, I am no longer in a hurry for an answer. I just realized that I can prove the global Frobenius theorem (every involutive distribution gives rise to a foliation) using only connectedness of the integral manifolds, not path-connectedness.