Let $B^{n+1}$ be the unit ball in the complex vector space $mathbb{C}^{n+1}$ with the standard Hermitian metric. Let ${Sigma}^n={partial}B^{n+1}=S^{2n+1}$ be the boundary sphere with the induced CR structure. Let f : ${Sigma}^n{hookrightarrow}{Sigma}^N$ be a local CR immersion. If N < 3n - 1, the asymptotic vectors of the CR second fundamental form of f at each point form a subspace of the CR(horizontal) tangent space of ${Sigma}^n$ of codimension at most 1. We study the higher order derivatives of this relation, and we show that a linearly full local CR immersion f : ${Sigma}^n{hookrightarrow}{Sigma}^N$, N $leq$ 3n-2, can only occur when N = n, 2n, or 2n + 1. As a consequence, it gives an extension of the classification of the rational proper holomorphic maps from $B^{n+1}$ to $B^{2n+2}$ by Hamada to the classification of the rational proper holomorphic maps from $B^{n+1}$ to $B^{3n+1}$.