Let $f: M \rightarrow M$ denote a diffeomorphism of a smooth manifold $M$. Let $p \in M$ be its hyperbolic fixed point with
stable and unstable manifolds $W_S$ and $W_U$ respectively. Assume that $W_S$
is a curve. Suppose that $W_U$ and
$W_S$ have a degenerate homoclinic crossing at a point $B\ne p$, i.e.,
they cross at $B$ tangentially with a finite order of contact.
It is shown that, subject to $C^1$-linearizability and certain conditions on
the invariant manifolds, a transverse homoclinic crossing will arise
arbitrarily close to $B$. This proves the existence of a horseshoe structure
arbitrarily close to $B$, and extends a similar planar
result of Homburg and Weiss .